Anal. Chem. 1986, 58,3148-3153
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(4) Mattheis, J. R.; Mitchell, G. W.; Spencer, R. D. I n New Directions in Molecular Luminescence;Eastwood, D., Ed.; ASTM: Philadelphia, PA, 1983; p 50. (5) Demas, J. N.; Keller, R. A. Anal. Chem. 1985,5 7 , 538. (6) Veselova, T. V.: Cherkasov, A . S.; Shirokov, V. I . Opt. SPectrosc. (fngl. Trans/.) 1970, 29, 617. ~. 1981,5, (7) Lakowicz, J. R.; Cherek, H J. Biochem. ~ i o p h y Methods
19
(8) Mousa, J. J.; Winefordner. J. D. Anal. Chem. 1974,4 6 , 1195. (9) Long, G. L.; Winefordner, J. D. Anal. Chem. 1983,5 5 , 712A.
RECEIVED for review April 14, 1986, Accepted August 4, 1986. This work was suPPo*d by the United States Army Research Office.
Minimization of Photofading and Drift- Induced Errors by Spectrum Scanning Strategy Sergio Cova* and Antonio Longoni Centro Elettronica Quantistica e Strumentazione Elettronica del CNR, Istituto di Fisica and Dipartimento di Elettronica, Politecnico di Milano, piazza Leonard0 da Vinci 32, Milano 20133, Italy Paolo A. Giordano and Isabel Freitas Centro Istochimica del C N R , Dipartimento di Biologia Animale, Uniuersitci di Pauia, piazza Botta 10, Pavia 27100, Italy
Sample photofading or other causes of slow drlft of the signal intensity can induce signlflcant dlstortlons in fluorescence spectra measured wlth wavelength scannlng apparatus. These distortions are analyzed and quantltatlvely assessed in terms of photofadlng rate and scanning time. For a glven measurement time, lt is shown to be advantageous to average spectra over fast repeated scans instead of a single slow scan. Furthermore, an up-down scannlng mode is shown to be preferable to an up-up mode. Experlmental data, abtalned on typical cytochemical samples, confirm the results of the analysls.
1. INTRODUCTION
Measurements of fluorescence spectra (emission or excitation) often have to be performed on fluorescent sources having nonconstant intensities. In spectrometers employing monochromators with a wavelength scanning device, this may imply significant distortions, as the spectra intensity at different wavelengths (A) is measured at different times. The problem is particularly relevant in cytochemical analysis with fluorescent staining techniques. The intensity of fluorescent emission from stained cells is usually not only weak but also decreasing with time because of photodecomposition effects ( I , 2), often denoted as photofading. A weak intensity requires a correspondingly long measurement time, T M , in order to enhance the signal-to-noise ratio (S/N) in the measurement. On the other hand, over a long T M the decrease of intensity due to photofading may often be significant. The question therefore arises of finding measurement procedures that minimize distortions without decreasing the S/N. A possible approach had been devised (1) that takes into account the observed dependence of the photofading rate on the effective excitation intensity. The photofading rate decreases with the intensity faster than linearly, so that it was suggested (i) to use a low effective excitation intensity (either by attenuating the source intensity or by using light with a narrow bandwidth around a wavelength shifted away from the maximum in the absorption spectrum) and (ii) to use a
short scanning time, such that negligible photodecomposition effects would be observed. Under such conditions, however, the available fluorescence intensity is often very low and the S / N is therefore unsatisfactory. The S / N may then be enhanced by averaging the spectrum over repeated scans, possibly exciting a different point of the sample in each scan (1). However, the total time taken by the measurement becomes in many cases undesirably long. The situation may be somewhat improved by exploiting the fact, reported in ref 3, that in conditions of short illumination periods, followed by recovery periods in darkness, a significant reduction of photofading effects is observed. If a light chopper is introduced in the excitation beam, it is then possible to use a higher excitation intensity and obtain a correspondingly higher fluorescence intensity. As choppers and lock-in amplifiers are commonly used for measurements of low light intensities, this approach is quite valuable. For a given S / N , however, the total time taken by the measurement is not much shorter, as the reduction in photodecomposition rate is not very high and half of the time (or more) has to be taken by the dark intervals. Therefore, the problem of performing in shorter times spectral measurements free from distortions due to photofading effects and with good S / N still deserves attention. The problem has been tackled here in the context of the development of a microspectrofluorometer for cytochemical research ( 4 ) . The starting point was the observation that (a) the obtainable S / N depends essentially on the total measurement time T M , not on the scanning procedure adopted within this time, provided the filtering type is suitable for scanning measurements with different scanning rates (see Appendix) and (b) the spectrum distortions are different for different scanning procedures. A reduction of photofadinginduced distortions may therefore be sought by looking for a suitable scanning strategy adoptable to the available spectrometric apparatus. The matter has to be quantitatively analyzed, and two intuitive considerations provide a useful guideline. First, the shorter the time taken by a scan, the lower the induced distortion. It is therefore advantageous to subdivide the available measurement time, T M , in various scans and to average the measurement over these scans. Second, if the fluorescent source intensity monotonically decreases
0003-2700/86/0358-3148$01.50/0 1986 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 58, NO. 14, DECEMBER 1986
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measurement. Of course, it is desirable to minimize the X dependence of the weighting w. Figure l b represents a measurement performed in a single slow scan from A = 0 to X = A,,,, lasting for time TM. For the sake of simplicity, let us suppose a linear relationship between wavelength position and scanning time. Therefore, and the the wavelength X is reached at a time t = TMh/X,,, light intensity is then decayed with respect to the beginning of the scan (time t = 0) by a factor f ( X ) = f(TmX/Xmax). The measured spectrum I,(X) is therefore distorted by the weighting function (drawn in Figure IC)
I ( A) ) = I, t h )f ( t 1
OO
TM
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t
ws(X)
tA
Flgure 1. For a case of fluorescence intensity decaying in time as sketched in a, the weighting function w (see text) in the measurement of the emission spectrum is drawn (onthe right) for three different types of wavelengths scanning (on the left), having the same total duration T,, namely: b and c a single slow scan: d and e two fast scans with equal sense (up-up); f and g two fast scans with opposite sense (up and down).
within a scan, the spectral positions that are measured later will be depressed. So, if the spectral range is scanned in the increasing A sense (up sense), the spectrum will be progressively depressed on the longer wavelength side. Conversely, if the scan starts from the upper spectral end, going in the opposite sense (down sense), the shorter wavelength side will be depressed. Therefore, some compensation of the spectrum shape distortions may be expected by alternating up and down scans in the averaging sequence. The approach may be expected to be valid also for compensating spectrum distortions induced by low variations of the signal intensity due to causes other than photodecomposition effects. Typical examples are distortions caused by a slow drift in the intensity of excitation lamps. In order to check the suitability of this approach to such cases, it is also necessary to take into account cases of rising intensity. It is worth stressing, however, that the technique is inherently limited to relatively slow variations of the intensity, so that, for instance, sudden jumps in the intensity of an excitation lamp cannot be compensated in this way.
= f(TMX/Xmax)
(3)
The intensity in the longer wavelength side is thus underestimated with respect to the short-wavelength side. In this measurement, the signal relative to each spectral position X is averaged over a time, ATM = T M A X, ,J ,X where AX is the spacing between the measured positions. Figure Id represents the case of two fast scans each having duration TMs = TM/2, so that the total duration is equal to the measurement time TMof the single slow scan previously described. For the sake of simplicity let us consider the time between successive scans negligible. Each scan therefore lasts TMs= TMf 2 and is performed upward (that is, in the sense of increasing X values). I t is to be stressed that the total averaged time, T M , for each X position is equal to that of the single slow scan, so that an equal signal-to-noise ratio results. The “weighting” function for the first and second scan are, respectively (see Figure le)
w~u(X) = f(TMX/2Xmax) = f(TM/2
W,U(X)
+ TMX/2Xmax)
(4) (5)
and the weighting function for the average of the two scans is WUU
=
1 ,[Wl“(X)
+ w,u(X)l
(6)
It is apparant that wuu(X) is less sloping that ws(X),as shown in Figure lc,e. Figure If concerns the case of two fast scans, each having duration T M S = TM/2, but performed in the opposite sense, the first upward and the second downward, with a total duration still equal to TM. The two weighting functions (see Figure lg) are, respectively wlU
= f(TMX/2Xmax)
(7)
The weighting function for the average is
2. ANALYSIS OF SPECTRUM DISTORTIONS Let the detected light intensity be decaying according to a law Z(X,t) = I&) f(t)
(1)
where f ( t ) ,which may not be exactly known, is sketched in Figure l a . As the various spectral positions are measured a t different times in a sequence depending on the measurement procedure, the measured spectral intensity Z,(X) is by a “weighting modified with respect to the original one lo@) function” w(X) (eq 2 ) , which depends on the features of the
The slope of wuD(X) is clearly less pronounced than that of Wuu(N. The distortion of the shape of the spectrum, due to the progressive underestimation of the spectral intensity with increasing A, can be characterized by the relative deviation between any two wavelengths XI and A,
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For an exponential decay law with a relaxation time TF, that is, with f ( t ) = exp(-t/TF), from eq 4, 5, and 6 it follows
and from eq 7 , 8, and 9
2
16’ :
0 L
0
+ u m (L
I t is readily seen that if the decay law can be approximated by a linear function over TMS, that is, if T M S
-=W X
2 As also graphically evident from Figure lg, this means that as far as f ( t ) is well approximated by a first-order expansion (linear function), the up-down scanning provides a practically perfect compensation of the distortion, while the up-up scanning implies a residual distortion, proportional to the wavelength interval. For other causes of intensity drift, characterized by a f ( t ) having a shape approximated over the scanning time TMS by functions different from exponentials, equations different from eq 11and 12 would be obtained. However, the conclusion that the up-down scanning provides a first-order compensation of the distortion is of general validity. Equations 11and 12 can be used to derive rational criteria for the selection of suitable operating modes in difficult cases, namely, those cases in which a sufficiently high signal-to-noise ratio is obtained only with a measurement time (T,) long enough to allow remarkable photodecomposition effects. This means cases with T M comparable to or not much shorter than TF,that is, for T F = FTM with F not much higher than unity. All measurements performed with different number of scans, Ns, but with the same total time TM= NsTMs, have the same signal-to-noise ratio, so that they can be compared on the basis of their maximum relative deviation D(O,X,,). Equations 11 and 12 can be readily extended to multiple scanning, as Duv applies to each scan and DUD to each couple of up and down scans. Therefore, by noting that TMs/TF= 1 / N s F , one has
Plots of DUUimax) and DUD(max) vs. N s are drawn in Figure 2. The superior effectiveness of the up-down scanning is quite evident; for instance, with F = 1,in order to limit D, at about 10% one has to subdivide in 10 up-up scans, while 1 up-down cycle (2 scans) is sufficient; for limiting D, at about 1%, 100 up-up scans are required, while 3 up-down cycles (6 scans) are sufficient. As mentioned in the introduction, it is also worth while to analyze the suitability of the technique for compensating distortions induced by other causes of intensity drift, typically drift in the excitation lamp. The main point is to verify the ability to cope also with a signal intensity that increases vs. time. This may be analyzed by approximating the intensity increase with a rising exponential; that is, by considering the
,\
, ,
10
100
Number o f scans N, Figure 2. Effectiveness of averaging over repeated up-up (subscript UU) and up-down (subscript UD) scanning in reducing spectral distortions due to photodecomposition of the sample. Measurements with given total time T , subdivided in N s scans; exponential intensity decay with rehxation time T, = FT,. Plots of the maximum relative deviation D, between the two spectral ends (A = 0 and A = Amx) are given; for clarity, continuous lines corresponding to eq 15 and 16 interpolate the discrete points.
results corresponding to f ( t ) = exp(t/TF). For the up-up scanning, instead of eq. 11 and 15 one obtains
D’uu(X1,X2)= 1 - exp
[ +--
(17)
Intuitively, the distortion is opposite to the case of photofading: the intensity a t longer wavelength is enhanced, as expressed by the negative sign of D‘uu. On the other hand, in the up-down scanning the change of sign in the exponential does not produce any change in the result: eq 12 and 16 apply also to this case. The plot of DUD(m,)in Figure 2 is still valid, while it is worth noting that for low values of N s the absolute value of D),’ is much higher than the corresponding value of DUU(max,. I t may therefore be concluded that the up-down scanning is definitely advantageous also in cases of the signal drifting toward higher intensity. 3. EXPERIMENTAL TESTS A microspectrofluorometer apparatus, described elsewhere ( 4 ) , was used in the experimental tests. A central digital unit controlled a stepping motor, driving the monochromator scan and the electronic measurement instrumentation. The relevant parameters in the scan were thus selectable and controllable with great flexibility. The intensity measurements were performed by single-photon counting, employing a multichannel pulse analyzer (MCA) in multiscaler mode. The MCA channel-scanner
ANALYTICAL
CHEMISTRY,
Wavelength
Table I. Procedure Used for Staining Frog Erythrocytes by Feulgen Reaction with Acriflavine-S02 reagents fixation hydrolysis staining wash
methanol-formalinacetic acid, 85:10:5 hydrochloric acid, 4 N acriflavine + SOz,8 X g/mL H20 + SO2 ethanol acidified with
VOL. 513, NO. 14, DECEMBER 1986
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I 8000
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6000 U
HCl
distilled water alcoholic scale xylol mounting medium Fluormount (E, Gurr, England)
b
dehydration
*
Room temDerature. *Brief rinsing. was driven synchronously with the monochromator scanner, so that a one-to-one correspondence between channel number and monochromator position was enforced. A prism monochromator was used; the selected wavelength was therefore related to the monochromator position with a nonlinear law, depending on the prism characteristics. The comparison between the experimental data and the results of the analysis in section 2 has therefore to be performed employing, instead of the wavelength scale, which is nonlinear with time, the actual geometric position in the scan, measured in steps and linearly related to time, as assumed in section 2. The maximum available scanning rate was 200 steps/s, corresponding to about 200 nm/s. A well-known type of cytochemical sample was used, namely, frog erythrocytes, stained by means of Feulgen reaction with Acriflavin-S02, as reported in Table I. Such samples have a well-known photofading behavior, quite suitable for these tests (1,2,4). The fluorescence emission intensity vs. time is not strictly exponential but may be approximated by a succession of exponentials with progressively increasing value of decay time Tp The observed photofading rate, that is, the value of 1/Tp, decreases faster than linearly with decreasing excitation intensity. It is thus possible to graduate the photodecomposition effects by graduating this intensity, and it is possible to obtain a measurement of the spectrum which is practically unaffected by photofading and can thus be used as a reference in the following tests by using the procedure suggested in ref 1 (see the Introduction). The sample also has a suitable excitation spectrum and a suitable shape of the emission spectrum, which has broad, unresolved peaks a t 550 and 600 nm, respectively. A xenon high-pressure arc lamp (XBO 75 W), filtered by a monochromator set at 450 nm, was used as the excitation source. The microscope worked in epiillumination with a FP 54X immersion objective; a TK 495 dichroic mirror was used to separate excitation and fluorescence. The raw collected spectral data are reported, without correction for the monochromator transmission and for the detecting photomultiplier spectral response. Therefore the results may be evaluated in terms of the detected light intensity, measured in photon counts per second, but may not be compared to corrected spectra. Two series of measurements were performed. Tests on the Stained Cell Spectrum. As shown in Figure 3a, a reference measurement of the emission spectrum between 480 and 700 nm was first performed a t low excitation intensity, with the procedure suggested in ref 1. The excitation intensity was then increased, so that the photofading time constant TFwas about 70 s, and measurements were performed by up-up and up-down scanning, with T m E 13 s per scan. The experimental results, depicted in Figure 3b,c are consistent with eq 11and 12. The up-down spectrum is practically equal to the reference spectrum. In particular, the value of the ratio of the two spectral peak amplitudes is equal to the reference value 1.12. In the u p u p spectrum the expected depression at longer wavelength is found; in particular, the measured value of the peak ratio is 1.07, as expected on the basis of eq 11. Tests on a Uniform Spectrum. The comparison between spectra of stained cells obtained in the presence of photofading
,
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-
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a'.
'.
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Channels
Figure 3. Emission spectra of the test samples measured with a scanning rate of 100 points per second (correspondingto about 150 nm/s): (a) reference spectrum, practically free from photodecomposition effects, obtained at low excitation intensity by the procedure described in ref 1 (average of 20 spectra, each measured with a single scan and illuminating a different location in the sample);(b) spectrum measured with up-up scanning at higher excitation intensity, corresponding to a photofading time constant rF = 70 s; (c) spectrum measured with up-down scanning at the same excitation level as in b.
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ANALYTICAL CHEMISTRY, VOL. 58, NO. 14, DECEMBER 1986
(see ref 4) was then disconnected, before performing the measurement of the spectrum. A lower excitation intensity was employed in this experiment,correspondingto a photofading time TFof about 150 s. Modulation of the excitation beam by a chopper wheel was introduced and lock-in detection was used to accurately measure the weak emission intensity (see ref 4). The experimental results are in agreement with eq 11-16, as shown by Figure 4. No distortion is observable in the up-down case and a linear sloping is visible in the up-up case. It is interesting to note that the value of the maximum deviation Dw(-)= 0.06 is consistent with results reported by other experimenters ( 3 ) ,denoting a remarkable reduction of photofading effects in conditions of short illumination periods, followed by recovery periods in darkness. In fact, a 6% value is given by eq 13 for a photofading time TF N 370 s.
50000
c v1
5
40000-
U
2 UP-UP scans
4. CONCLUSION I Averaging of fast repeated wavelength scans reduces the
30000~
20
40
60
80
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Channeis
50000 / b
c
2
40000
u
i !
1 UP-DOWN cycle
Channels Figure 4. Test of the compensation of the distortion due to photofading on an experimental simulated uniform spectrum (see text). Emission from the test samples at 550 nm (first peak in figure 3). Measurements with chopper modulation at 250 Hz and lock-in detection; 100 chopper cycles per channel per scan with half-cycle gate-time TG = 2 ms; and scanning rate 2.5 channels per second: (a) 2 up-up scans: (b) 1 up-down cycle (2 scans). Note the expanded vertical scale.
and in reference conditions, respectively, gives significant quantitative indications about the effectiveness of the correction technique. However, with such a test it is not possible in practice to obtain with good accuracy the shape of the function describing the relative deviation factor D vs. wavelength A. In principle, by taking the ratio of the corrected spectrum to the corresponding reference spectrum, normalized to the initial amplitude, one should obtain the function [l - D(O,A)]. In practice, however, various reasons make such a result not very accurate and reliable. Differences between two spectra taken from different locations in the cytochemical sample may be due to biological variations or variations in the stain uptake; on the other hand, if the two spectra are taken from the same location one after the other, variations in the second measurement may be suspected to be caused by the first irradiation. Furthermore, the ratio of the two spectra is obviously affected by high statistical dispersion at wavelengths where the spectral amplitude is small. An accurate experimental determination of [ I - D(O,X)]vs. X would require a situation where the differences between the reference and the actual measurement are caused only by photodecomposition effects and where the spectrum amplitude is uniform. Such a uniform spectrum was not availablefrom cytochemical samples;however, an experimental procedure was devised that produced a situation equivalent to it. In fact, for the test purposes, a uniform spectrum subject to distortion only by photofading effects can be readily simulated, by keeping constant the position of the monochromator while scanning the scale of the electronic measurement apparatus, that is, the channels of the MCA in our case. Therefore, the emission monochromator was first set at the first spectral peak (550 nm); the line carrying the step command to the motor driver circuit
distortions of the measured spectrum produced by sample photodecomposition or other causes of slow drift of the signal intensity (drifting excitation lamps, etc.). By use of the updown mode in the repeated scans, an effective compensation of errors induced by photofading or other drift is introduced. The analysis points out and experiments do confirm that adopting up-down scanning may be considered equivalent to an increase by more than 1 order of magnitude in the rate of scanning in the up-up mode, as far as a slowly varying intensity is concerned. In cytochemical spectrofluorometric measurements, the results of the analysis performed in section 2, and in particular eq 11-16, can quite generally be used to select the measurement procedure in the presence of photodecomposition effects. Although the photofading behavior of stained cells is generally not a simple exponential over all the measurement time ( I , 2), the actual decay can be well fitted by simple exponential functions exp(-t/ TF)over limited time intervals, of the order of TFor shorter. Usually, the fitting value of TF increases as time elapses, more rapidly a t the beginning and then progressively slower ( I , 2). Equation 11thus applies to each scan and eq 12 to each couple of scans, provided TMs < TF, with progressively increasing values of TF. Therefore eq 15 and 16 can still be used; a conservative estimate of D,, will be obtained by using the F value corresponding to the beginning of the decay, and more closely approximate values of D,, will be estimated by using an appropriate average value of F.
ACKNOWLEDGMENT The authors are indebted to N. Omenetto of CCR Ispra, Italy, for helpful discussions and for his critical review of this paper. APPENDIX To be considered suitable for measurements with a scanning spectrometer, the filtering acting on the detector signal and noise should satisfy a basic requirement: at any scanning rate, it must exploit the information contained in all the time interval available for each spectrum point and only in that interval. In fact, if the filter weighting ( 5 ) extends over a longer interval, the measured spectrum shape will be altered and the resolution will be correspondingly degraded; if it extends over a shorter interval, the S / N will not reach the available value. For a given filtering type, the filter parameters must therefore be changed as the scanning rate is changed. This is automatically obtained with filtering given by a gated integration over the time devoted to each spectral point in the scan. In a single-photon counting apparatus, this kind of filtering is normally implemented. I t may also be implemented with analog detectors, by using an analog gated integrator followed by an analog-to-digital converter (ADC). It may be more or less closely approximated by using analog time-invariant averaging filters (6) followed by a sampler and ADC, provided the filtering parameters are properly read-
Anal. Chem. 1986, 58, 3153-3158
justed when the scan rate is changed.
LITERATURE CITED (1) Prenna, G.; Mazzini, G.; Cova. S. Histochem. J . 1974, 6 , 259. (2) Cova, S.;Prenna, G.; Mazzini, G. Histochem. J . 1974, 6, 279. (3) Rundquist, I.; Enerback, L. Histochemlstty 1978, 4 7 , 79. (4) Longoni, A.: Cova, S.; Giordano, P.; Freitas, I.Internal Report 86-014, Dipartimento di Elettronica, Politecnico di Milano; also to be submitted for publication.
3153
(5) Cova, S.; Longoni, A. I n Analytical Laser Spectroscopy; Omenetto, N., Ed.; Wiley: New York, 1979; Chapter 7. (6) Voigtman, E.; and Winefordner, J. Rev. Sci. Znstrum. 1986, 57, 957.
RECEIVED for review April 1, 1986. Accepted August 7, 1986. Work supported by Consiglio Nazionale delle Richerche and Minister0 Pubblica Istruzione.
Extension of Accessible First-Order Rate Constants and Accurate Dead-Time Determinations for Stopped-Flow Spectroscopy Peter N. Dickson and Dale W. Margerum* Department of Chemistry, Purdue University, West Lafayette, Indiana 47907
Systematlc errors are encountered In the measurement of rapid reactions with stopped-flow instruments due to Ilmltatlons of mixing rates. Observed flrst-order rate constants ( k - ) can be smaller than the true rate constant ( k , ) because of the Influence of the mixing rate constant (km,,)on the reaction profile after the flow has stopped. A data treatment to resolve k , from k,, Is given based on the rel/km,x,Thls permlts pseudolationship l/kow = l/k, first-order rate constants (&,) in the range of 100-2000 s-’ to be determlned wlth good accuracy by use of a commonly used stopped-flow Instrument. Larger k , values (>3000 s-‘) can be resolved from instruments with larger k,, values and shorter dead times. A slmple phenomenological determlnatlon of dead time (td) is also descrlbed based on !d = In (hAp‘*/hA-)/kobsd, where AAw* and AAow are the predicted and observed absorbance changes in the stoppedflow cell. A chemlcal calibration system is recommended.
+
Stopped-flow spectroscopy is a widely used method to measure reaction rates. Frequently, large first-order and second-order rate constants are reported without consideration of the systematic errors that are introduced as a result of the physical process. Limitations in the ability of stopped-flow instruments to provide useful information about fast reactions are governed by the instrument’s mixing rate and dead time as well as the magnitude of the absorbance change for the reactions. Some of these limitations have been discussed by Hiromi ( I , 2) and by Rorabacher (3), but the problem of incomplete mixing has not been addressed. The physical mixing of solutions is a rate process that must precede the chemical reaction. The degree of mixing at the time the flow stops affects the magnitude of the chemical reaction rate that can be measured. In this work we show how to correct rapid first-order reactions for mixing effects. The dead time may be defined as the time during which the physical and chemical processes initiated by mixing proceed without detection. Dead time is a result of the physical separation of the mixer and the detector; this causes a temporal separation of reaction initiation and data acquisition. The effect of this temporal separation is an initial 0003-2700/86/0358-3 153$0 1.50/0
absorbance jump in the data (increase or decrease) and a corresponding decrease of the observed absorbance change (i.e., a diminished signal change for the overall reaction). The signal change becomes so small for very fast reactions that it is no longer distinguishable from the background noise. One solution to this problem is to increase the expected signal change until a measurable fraction remains after the dead time. Passive filtering (“time constant”) and ensemble averaging help to improve signal-to-noise ratios (S/N). Large total signal changes, absorbance offsetting, filtering, and ensemble averaging permit the measurement of large first-order rate constants with conventional stopped-flow instrumentation. However, these observed rate constants, kobd, are often smaller than the true rate constant, k,. Pseudo-first-order rate constants measured by stopped-flow instruments may show significant deviation from ideal behavior when the rate of the chemical reaction becomes fast. The rate constant at which this deviation becomes important depends upon the efficiency of the mixing process. Sometimes this deviation is used to set an upper limit a t which an instrument is useful (1-4). These upper limits have varied from 200 to 2000 s-l and depend on the instrument and the evaluation procedure used. We have previously indicated ( 4 ) that precautions must be taken with the Durrum-Dionex instrument (with a 20-mm observation tube) when measuring rate constants larger than 230 s-l and that significant deviations occur above 350 s-’. In the present work we show that small corrections are needed even for rate constants of 100 s-l, However, given a sufficiently large signal-to-noise ratio, much larger first-order rate constants can be observed with this instrument. These values do not accurately measure the rate of the chemical reaction because the mixing rate has a measurable contribution in very fast kinetic processes. Characterizing the effect of the mixing rate constant, kmi,, permits accurate measurement of chemical reaction rate constants where koklsd is a combination of k , and k,,,. For example, with one instrument, the observed rate constant is 1100 s-l, whereas the true rate constant is 1820 s-l. In some cases the magnitude of the dead time of a stopped-flow instrument is important. The analysis of secondorder kinetics (and other rate processes that are not first order or zero order) often requires the use of actual time in the analysis. Typically, the experimental time axis is offset from 0 1986 American Chemical Society
