14914
J. Phys. Chem. 1996, 100, 14914-14921
Acid-Base Equilibria Involved in Secondary Reactions Following the 4-Carboxybenzophenone Sensitized Photooxidation of Methionylglycine in Aqueous Solution. Spectral and Time Resolution of the Decaying (S∴N)+ Radical Cation Gordon L. Hug,*,† Bronislaw Marciniak,†,‡ and Krzysztof Bobrowski†,§ Radiation Laboratory, UniVersity of Notre Dame, Notre Dame, Indiana 46556, Faculty of Chemistry, A. Mickiewicz UniVersity, 60-780 Poznan, Poland, Institute of Nuclear Chemistry and Technology, 03-195 Warsaw, Poland, and Institute of Biochemistry and Biophysics, Polish Academy of Sciences, 02-106 Warsaw, Poland ReceiVed: January 29, 1996; In Final Form: June 26, 1996X
A radical cation with an intramolecular sulfur-nitrogen bond was formed in the photoinitiated transfer of an electron from the sulfur atom of the dipeptide Met-Gly to 4-carboxybenzophenone in its triplet state. The sulfur-nitrogen coupling involved two-center, three-electron bonds. The kinetics of the reactions of these radical cations, which were initiated by a laser flash, were followed over time. The spectrum of the radical cation had to be extracted from the interference of other absorbing transients. The principal method of implementing the spectral resolutions was accomplished through a multiple linear regression technique. This spectral analysis was repeated for numerous time windows during the lifetime of the transients’ decays. The resulting concentrations of the transients were consistent with an independent factor analysis. It was found that the decay of the radical cations was multiexponential and that the decay varied with pH. A simplified reaction scheme was proposed whereby the absorbing radical cations can alternatively decay by an irreversible channel or react reversibly with OH-. Rate constants for the three elementary reactions of this scheme were determined from an analysis of the decay of the concentration of the radical cations. In addition, the equilibrium constant for the reversible reaction was determined by two separate procedures. In one, the equilibrium constant was calculated from the approximate linear [OH-] dependence of the faster of the two rate constants (in the observed two-exponential decay). The second determination of the equilibrium constant was made from the preexponentials of the empirical two-exponential fits. It was shown how these preexponentials were related to the equilibrium concentrations of the species involved in the equilibrium.
Introduction
CHART 1
Oxidative damage to proteins has been intensively investigated recently as a source of disease states in biological species.1,2 The metabolism of reactive species has wide interest in medicine,3 physiology,4 and radiation biology.5 One of the sites attacked by environmental and internal oxidative agents is the sulfur atom in the side chains of certain amino acid residues in proteins. The fate of the reactive species formed by an initial attack at a sulfur atom is of interest in following the course of the reaction to the ultimate damage site and in understanding possible repair mechanisms. There has been much previous work done on sulfur-containing amino acids, in particular methionine6-13 and methioninecontaining peptides,14-17 where the emphasis was on the study of the fast initial reaction following oxidation. Pulse radiolysis and flash photolysis have been used not only to create the relevant radicals and radical ions but also to resolve the fast initial reactions of these species. Pulse radiolysis techniques have the advantage of generating radicals and radicals ions without much interference from absorbing substrates. However, oxidizing species, even strong ones such as •OH, can have the complication that they are likely to form radical cations from sulfur through the intermediary of an adduct with the oxidizing species, such as a >S•OH adduct.18,19 The use of flash * To whom correspondence should be addressed. † University of Notre Dame. ‡ A. Mickiewicz University. § Institute of Nuclear Chemistry and Technology and Institute of Biochemistry and Biophysics, Polish Academy of Sciences. X Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)00367-X CCC: $12.00
photolysis, on the other hand, also has complications because the absorbing substrates and other absorbing radicals can interfere with attempts to follow the species of interest. However, the advantage of flash photolysis is that radical cations can be formed directly either by photoionization20 or by photosensitized electron transfer from a substrate.11 The current work shows how, by extending previously introduced analytical methods, the limitations of the flash photolysis technique for the study of radicals and radical ions can be overcome. We extend our previous work12 of using a multiple linear regression analysis21 to handle the time resolution of a complicated kinetics scheme involving a dipeptide with a methionine at the N-terminal (see Chart 1). A triplet aromatic ketone is used to accept an electron from the sulfur atom leaving © 1996 American Chemical Society
Spectral and Time Resolution of Decaying (S∴N)+ a radical cation site on the sulfur. It has previously been shown that the unpaired electron on this site can form a two-center, three-electron bond9 with one of the lone pairs of the nitrogen at the N-terminal of the peptide. The product is a relatively stable species14 with a five-membered ring (S∴N)+ (see Chart 1). This species absorbs around 400 nm. However, since aromatic ketones were used to sensitize the creation of the radical cation, the absorptions of the resulting ketyl radicals and radical anions interfered with the absorption of the radical cation.17 Preliminary work showed that by using spectral resolutions at selected time delays after the laser pulse it was possible to extract the spectrum and hence the concentration of the (S∴N)+ species.17,22 This seemed to indicate that the decay pattern of the (S∴N)+ was a complex function of the pH. The current work is an attempt to unravel both the time and spectral resolution of the (S∴N)+ species. The results, using multiple regression analysis for spectral resolution, were supported by the related, but independent, methodology of factor analysis.23 Experimental Section Materials. 4-Carboxybenzophenone (Aldrich) (CB) and the peptide methionylglycine (Bachem) (Met-Gly), with both amino acids in their L-isomeric configurations, were of the purest commercially available grade and were used as received. See Chart 1 for the structure of Met-Gly. Water was purified by a Millipore Milli-Q system. Laser Flash Photolysis Experiments. The nanosecond laser flash photolysis setup has been described in detail elsewhere.24 Laser excitation at 337.1 nm from a Laser Photonics PRA/Model UV-24 nitrogen laser (operated at 3 mJ, pulse width ≈ 8 ns) was at a right angle with respect to the monitoring light beam. The source of the monitoring beam was a 1 kW pulsed xenon lamp.25 The transient absorbances at preselected wavelengths were monitored by a detection system consisting of a double monochromator (Instruments SA, model H10) and a photomultiplier tube. The signal from the photomultiplier was processed by a 7912 AD Tektronix transient digitizer controlled by an LSI 11/2 microprocessor. Cutoff filters were used to avoid spurious response from second-order scattering off of the monochromator gratings. Experiments were performed in rectangular quartz cells (0.5 × 1 cm) with an optical path length of 0.5 cm for the monitoring beam. At each pH, 25 or 50 mL of solution was used for a complete series of traces. The solution was flowed, and kinetic traces were taken at 10 nm intervals, usually between 360 and 720 nm. From the wavelength-dependent traces, the time-dependent transient spectra could be constructed. The concentration of CB was usually ≈2 × 10-3 M in most experiments, but some experiments had as much a three times this concentration. The concentrations of Met-Gly were 0.02 M for all experiments. At this concentration, quenching of the triplet of CB was virtually complete (>99.5%). Typically 5-7 laser shots were averaged for each kinetic trace. The pH’s of the solutions were adjusted by adding sodium hydroxide. The pH’s of the solutions were measured with an Orion Research Model 811 pH meter with a Ross Combination electrode Model 81-02. No buffers were used. All solutions were deoxygenated by bubbling high-purity argon through them. Met-Gly was only added to the solutions just before the experiments were done because the pH was seen to vary slowly over several hours. The full set (37 from 360 to 720 nm) of kinetic traces took approximately 1.5 h to collect. Pulse Radiolysis Experiments. Pulse radiolysis experiments were performed with 1 ns pulses of high-energy electrons from
J. Phys. Chem., Vol. 100, No. 36, 1996 14915
Figure 1. Example of resolution of spectra: end-of-flash spectrum of aqueous solution of 2 mM CB and 20 mM Met-Gly at pH 10.06.
the Notre Dame Titan Beta Model TBS-8/16-1S electron linear accelerator. Absorbed doses were on the order of 4-6 Gy (1 Gy ) 1 J/kg). N2O-saturated 10-2 M solutions of potassium thiocyanate were used as the dosimeter, taking a radiation chemical yield of 6.13 and a molar extinction coefficient of 7580 M-1 cm-1. A description of the pulse radiolysis setup, data collection system, and details of dosimetry has been reported elsewhere.25,26 The data acquisition subsystem has been updated27 and includes a Spex 270M monochromator, a LeCroy 7200A transient digitizer, and a Gateway 2000 (486/66 MHz) computer. The software was written within LabWindows of National Instruments.27 The experiments were carried out with continuously flowing solutions. Results and Discussion 1. Resolution of Transient Spectra. Time-resolved spectra were taken following the laser flash. A flow system was used, and individual kinetic traces were collected at 10 nm intervals. The time-resolved spectra were generated from these sets of traces, choosing appropriate time delays and appropriate time windows. Spectra, so generated, were resolved into component species by a linear regression technique21 of the form n
∆A(λj) ) ∑i(λj)ai
(1)
i)1
where ∆A(λj) is the observed absorbance change of the composite spectrum and i(λj) is the molar absorption coefficient of the ith species at the jth wavelength of observation. The linear regression coefficients correspond to cil (where ci is the concentration of the ith transient and l is the optical path length of the monitoring light). Further details of this method have been described elsewhere.12 The component spectra {i(λj)} of the transients were calculated. An example of a resolution of three transients is shown in Figure 1 for Met-Gly at pH 10.06. 2. Time Resolution of Transients. With the spectral resolution technique described above, it is possible to extract the absorption spectra of transient species that are partially hidden by the absorption of other transients. One way to study the time dependence of the (S∴N)+ radical cation is to do a spectral resolution at each point in time. This can be quite time consuming, but it has led to consistent results. We chose to employ this method in this paper. However, we did check our results with the more comprehensive method of factor analysis23 (see below).
14916 J. Phys. Chem., Vol. 100, No. 36, 1996
Figure 2. Example of the time resolution of the transient absorption spectra of an aqueous solution of 2 mM CB and 20 mM Met-Gly at pH 10.06. The three concentrations at each point in time comes from a separate spectral resolution, similar to that in Figure 1.
Figure 3. Decay of (S∴N)+ and CBH• at pH 6.72 for the purpose of measuring the disappearance of (S∴N)+ in the absence of OH-. Concentration profiles of the radicals were obtained from kinetic traces at 10 nm intervals between 360 and 720 nm in a manner analogous to that in Figure 2. Inset: traditional plot for second-order processes. The resulting 2k ) 1.6 × 109 M-1 s-1.
Some typical results are shown in Figure 2 for the decay of the concentrations of three intermediates. The data were obtained from a series of kinetic traces each at a different wavelength. The spectral resolution at each point in time was done independently of the spectral resolutions at the other time points. However, it can be seen that the concentrations vary in time in a reasonably continuous fashion which gives some credence to the plausibility to the method. The CB•- radical is persistent on the 6 µs time scale, whereas the (S∴N)+ and (S∴S)+ radical cations decay on this time scale. Another example of this method is shown in Figure 3 on a 150 µs time scale. The (S∴N)+ radical cations decay according to a first-order law with a rate constant of 2 × 104 s-1 or a lifetime of 50 µs. This lifetime will be used in the analysis below as the lifetime of (S∴N)+ in the absence of OH-. The ketyl radical CBH• decays by a second-order law with 2k ) 1.6 × 109 M-1 s-1. This rate constant was determined in the usual way from the plot of 1/[CBH•] vs time as shown in the inset of Figure 3. The pKa of CBH• was previously determined to be 8.2.28 As the example in Figure 2 shows, when the solutions were basic, the decay of the (S∴N)+ radicals was not, in general, single exponential. This decay and other (S∴N)+ decays that were found to be nonexponential in basic solutions are summarized in Figure 4. In Figure 4 the decays are all normalized to the same initial concentration of (S∴N)+. In the pH range above 10.6, the long-time tail in the decay of the concentration
Hug et al.
Figure 4. Comparison of the decays of (S∴N)+ at various pH’s. Each point in time, at each pH, involved a separate spectral resolution similar to that in Figure 1. Each time/pH point required a series of kinetic traces, each trace at a different wavelength. Concentrations were all normalized to the initial concentrations at each pH.
Figure 5. Two-exponential fit of the decay of the (S∴N)+ concentration at pH 10.58.
TABLE 1: Rate Constants of (S∴N)+ Decay (Fast Component) for Various pH’s ([CB] ) 2 mM, [Met-Gly] ) 20 mM, pH Adjusted with NaOH) pH
[OH-] (M)
τSN+ (ns)
kSN+ × 10-6 s-1
11.10 10.96 10.86 10.64 10.58 10.06 10.06 9.62
1.26 × 10 0.91 × 10-3 0.72 × 10-3 0.44 × 10-3 0.38 × 10-3 0.11 × 10-3 0.11 × 10-3 0.042 × 10-3
140 110 170 250 300 640 600 700
7.05a 8.98b 5.95b 4.00b 3.35a,c 1.56a,d 1.67a,e 1.42a
-3
a From a two-exponential fit. b One-exponential decay. c [CB] ) 6 µM. d 6 µs time scale. e 3 µs time scale.
of (S∴N)+ disappeared as a general rule. Even at pH 11.1 where a long-time component is observable (see Figure 4), this component is quite weak. Each of the (S∴N)+ decays in Figure 4 can be well fit to separate two-exponential functions. An example of a twoexponential fit of the (S∴N)+ decay at pH 10.58 is given in Figure 5. From Figure 4 it can be seen that the initial decays of (S∴N)+ appear to get faster as the pH of the solutions increases. This is reflected in the rate constants of the fast component of the two-exponential decays. These rate constants are listed in Table 1 as kSN+. 3. Kinetic Scheme. A plausible explanation for the increase in kSN+ as the OH- increases (see Table 1) is that (S∴N)+ is disappearing due to a reaction with OH-. A similar influence
Spectral and Time Resolution of Decaying (S∴N)+
J. Phys. Chem., Vol. 100, No. 36, 1996 14917
SCHEME 1
Figure 6. Linear least-squares fit of the fast rate constants of (S∴N)+ decay vs [OH-] as expected from eq A8.
of base on sulfur-centered radical cation decays was found in the benzophenone-sensitized photooxidation of 1,5-dithiacyclooctane.29 This observation was rationalized in terms of a neutralization reaction either of >S•+ radical cations and/or (S∴S)+ by hydroxide ions. A possible mechanism that could account for the twoexponential decays and the increase in the plateau value of [(S∴N)+] with the decrease in pH is the existence of an equilibrium. This equilibrium is hypothesized to exist between (S∴N)+ and some unspecified intermediate (symbolized by SNOH) formed by the reaction of (S∴N)+ and OH-. The simplest such mechanism is given in Scheme 1. However in an earlier work a more complete reaction scheme was presented.13 In the more extensive scheme there were two separate equilibria involving (S∴N)+ and OH-. The two intermediates involved in these two equilibria were a >S•-OH species and an analogous OH adduct at the nitrogen center. The involvement of an >S•-OH species in the equilibrium can be checked by alternate generation of the transient (see Time Resolution of Transients). The OH adduct to the nitrogen seems rather unlikely because it would involve a hypervalent nitrogen. However, provided that the equilibrium between the closed form (S∴N)+ and an open-chain nitrogen-centered radical cation is fast, the acid-base behavior could be rationalized within our scheme as a reaction of this open-chain radical cation and OH-, leading to a deprotonation at the nitrogen. The symbol SNOH will collectively represent these three possible species on the right-hand side of the equilibrium in Scheme 1. The kinetic equations for Scheme 1 can be solved by routine methods,30 and the solution is given in the Appendix for reference. There are three elementary rate constants that are required to characterize the scheme. For one of them, k3, it seems reasonable to use the rate constant (k3 ) 2 × 104 s-1) already determined (see Figure 3) at pH 6.72 for the decay of (S∴N)+ in the absence of OH-. It can be seen from eqs A1A4 that the empirical rate constants and preexponential factors are relatively complicated functions of the three elementary rate constants and [OH-]. However, since the largest [OH-] used in the experiments was about 1 mM, the tentative approximation was made to expand the empirical rate constants and the preexponential factors of the two-exponential function about [OH-] ) 0. Care must be taken to keep track of the signs when the square roots are expanded. Two cases result: one for k2 > k3 with eqs A5-A8 and one for the case k3 > k2 with A9A12. By simulating the decay of (S∴N)+ with trial-and-error choices for k1, k2, and k3, it was clear that the case of k2 > k3 had to be the operative one for these data. In the other case of k3 > k2, the simulated decay of (S∴N)+ always rapidly decayed,
leaving little or no long-time concentration of (S∴N)+. Physically this is because the k2 must be slow enough so that the OH adduct (SNOH in Scheme 1) can act as a storage for (S∴N)+ without (S∴N)+ decaying directly. However, if k3 is larger than k2, then as soon as (S∴N)+ forms, it either decays or is sent back to the OH adduct. No matter how fast the adduct forms, the kinetics of the scheme ultimately become a competition between k2 and k3. Therefore, to accommodate the data, we considered only the approximations in eqs A5-A8 for the case of k2 > k3. The most reliable parameters from the two-exponential decays are the fast empirical rate constants (kfast) because the time scales were chosen to get accurate determinations of the early decay features. The approximation in eq A8 was used to interpret the plot of kfast vs [OH-]. The plot is given in Figure 6 taking values from Table 1. A linear least-squares fit gave a slope of (6.0 ( 1.0) × 103 M-1 µs-1 and an intercept of 1.3 ( 0.7 µs-1. From eq A8, it can be seen that the intercept is equal to k2 ) (1.3 × 106 s-1) and that the slope is k1k2/(k2 - k3). Since k2 and k3 have been determined, k1 can be calculated from the slope to be 6.0 × 109 M-1 s-1, which is reasonable for a neutralization reaction in water. With these values for the three elementary rate constants, it is possible to go back and test the approximations A5-A8 in the pH range under consideration. Decay curves using both the approximate eqs A5-A8 and the exact eqs A1-A4 were computed for a series of pH values. Only at the lowest pH of 9 did the exact and approximate simulated decay curves match. At higher pH values the exact and approximate simulations diverged. In the higher range of pH, the approximate simulation disagreed so much as to give a growth and decay pattern, which was an indication that the approximations for the preexponential factors in eqs A5 and A6 had broken down. The slow rate constant in eq A7 also failed to agree with the exact rate constant, -φ+ in the high-pH range. On the other hand, the approximate rate constant from eq A8 agreed within 5% of the exact -φ- even at the highest pH (of 11) used in the experimental measurements. This can be taken as a justification for the use of eq A8 and Figure 6 in the determinations of the elementary rate constants k1 and k2. The complete reaction scheme13 indicates that the OH adducts to the sulfur and the nitrogen (symbolically SNOH in Scheme 1) can both decay, sometimes even with further OH- participation. Without any additional OH- participation, the inclusion of a k4 for the decay of SNOH in Scheme 1 would not change eq A8 by much. However, the intercept would now be k2 + k4, and the slope could only determine the product of k1 and k2. To determine all four elementary rate constants in this extension of Scheme 1, one of the other expansions analogous to eqs A5-
14918 J. Phys. Chem., Vol. 100, No. 36, 1996
Hug et al. TABLE 2: Determination of K from Equilibrium Concentration of Transients pH
[OH-] (mM)
[SN+]eq (µM)a
[SN+]0 (µM)b
[SNOH]eq (µM)c
11.10 10.58e 10.06f 10.06g 9.62
1.26 0.38 0.11 0.11 0.042
0.53 7.94 3.80 3.14 3.06
5.39 21.3 6.78 6.30 4.66
4.86 13.3 2.98 3.16 1.60
K (M-1)d 7300 4400 7100 9100 12000 8000 ( 2800
a
From preexponential factors to slow component in two-exponential fits. b Sum of preexponential factors in two-exponential fits. c From preexponential factors to fast component in two-exponential fits. d From eq 4. e [CB] ) 6 mM, but at all other pH’s, [CB] ) 2 mM. f For 6 µs time scale. g For 3 µs time scale.
Figure 7. Resolution of the spectral components in the transient absorption spectra obtained in the pulse radiolysis of N2O-saturated aqueous solution of Met-Gly (10 mM) at pH ) 9.6 taken (a) 20 ns, (b) 35 ns, and (c) 100 ns after electron pulse. (d) Growth and decay of >S•-OH intermediate and growth of (S∴N)+ intermediate obtained by the resolution of spectral components in the transient absorption spectra taken at various time delays.
A7 would have to be valid. Given the results of the simulations on the exact and approximate parameters from Scheme 1 itself, it is unlikely that the analogous equations to eqs A5-A7 would fare any better than eqs A5-A7 themselves. 4. Time Resolution of Transients in Pulse Radiolysis. To determine whether the hydroxy sulfuranyl radical, >S•-OH, is the SNOH species in Scheme 1, complementary pulse radiolysis experiments were performed at pH 9.6. In these experiments, the appropriate hydroxy sulfuranyl radicals were generated (Scheme 1) and their transformation into (S∴N)+ radical cations was monitored. The composite spectra taken up to 100 ns after the electron pulse are reminiscent of the (S∴N)+ radical cation, but they have shoulders at shorter wavelengths that make them broader than the spectrum of the (S∴N)+ radical cation (Figure 7a-c). This spectral feature is particularly prominent in the spectrum recorded 20 ns after the pulse (Figure 7a). The short wavelength shoulder is in the region of 330-340 nm, the absorption maximum of the hydroxy sulfuranyl radical (>S•-OH). These radicals were identified and well characterized in pulse radiolysis experiments on alkyl sulfide18,31 and substituted alkyl sulfide19,32 systems under conditions that allowed only the isolated transient. The (S∴N)+ radical cations were previously identified in pulse radiolysis experiments on Met-Gly14,17 and methionine ethyl ester.9 By resolving the transient spectra for a sequence of time windows, it was possible to separate the concentration profile of the (S∴N)+ radical cation from that of the hydroxy sulfuranyl radical (>S•-OH). The concentrations of these two radicals are plotted in Figure 7d as a function of time. The concentration profile of the hydroxy sulfuranyl radical has been simulated quantitatively as the superposition of a pseudo-first-order buildup (k ) 7.0 × 107 s-1) and a first-order decay (kd ) 4.7 × 107 s-1), whereas the concentration profile of the (S∴N)+ radical cation followed a first-order buildup (kb ) 4.8 × 107 s-1). Since the values of kd and kb are similar, this confirms direct transformation of the >S•-OH radical into the (S∴N)+ radical cation. However, such a fast reaction ruled out an involvement of the >S•-OH radical in the acid-base equilibrium with the (S∴N)+ radical cation since the calculated reverse rate constant k2 (Scheme 1) is 1 order of magnitude lower (k2 ) 1.3 × 106 s-1).
5. Equilibrium Constant. From the values of k1 and k2 determined from the empirical rate constant of the fast component of the (S∴N)+ decay, the equilibrium constant for the equilibrium (S∴N)+ + OH- a SNOH in Scheme 1 was determined to be 4600 ( 2600 M-1. The standard deviation was computed by the usual propagation of errors formula21 for division of k1 by k2. Another way to determine the equilibrium constant is from the equilibrium concentrations by
K)
[SNOH]eq
(2)
[(S∴N)+]eq[OH-]eq
If any irreversible decays of both species (S∴N)+ and SNOH could be totally neglected so that the two species only interconverted, then [(S∴N)+]eq ) [(S∴N)+]∞ and [SNOH]eq ) [(S∴N)+]0 - [(S∴N)+]∞. The subscripts 0 and ∞ refer to the initial concentrations and the concentrations at infinite time, respectively. In a two-exponential fit to such a hypothetical approach to equilibrium, the equilibrium concentrations of [(S∴N)+]eq and [SNOH]eq would come from the preexponential factors of the slow (infinitely slow for no irreversible decay) and fast components, respectively, in the decay of [(S∴N)+]. The relationship of an empirical two-exponential fit to a hypothetical single-exponential approach to equilibrium suggests that it might be possible to determine the equilibrium constant from the preexponential factors of the two-exponential fitting parameters. From the elementary rate constants above, the condition k3 , (k1[OH-] + k2) holds because k3/(k1[OH-] + k2) is less than 1/70 for all OH- concentrations. The ratio of the preexponential factors (see eq A2) can be expanded about k3/(k1[OH-] + k2) ) 0. After some computation it can be shown that
[SN+][SN+]+
)
(
)
k1[OH-] 2k3 1+ + ... k2 k1[OH-] + k2
(3)
Therefore for relatively slow irreversible decay, the equilibrium constant can be calculated using the preexponential factors from the two-exponential fits as
K≈
[SN+][SN+]+[OH-]
(4)
where [SN+]- and [SN+]+ are the preexponential factors of the fast and slow components, respectively, of the two-exponential fits. The values of the equilibrium constant computed from this approximate formula are given in Table 2. It can be seen that
Spectral and Time Resolution of Decaying (S∴N)+
J. Phys. Chem., Vol. 100, No. 36, 1996 14919
this equilibrium constant is about 1.7 times K computed from the elementary rate constants, which were, in turn, determined from the linear [OH-] dependence of the fast rate constant as in Figure 6. However, there are large uncertainties in both determinations. 6. Factor Analysis. A factor analysis23,33 was done in order to serve as an independent check on the spectral-resolution methodology. As an example, the data for pH 10.06, coming from a set of 37 kinetic traces on a 3 µs time scale, were analyzed both by the spectral-resolution method described above and by factor analysis. The wavelengths of the monitoring light for these traces varied between 360 and 720 nm. As can be seen below, the factor analysis gave roughly the same results as the spectral analysis taken time point by time point. Factor analysis is a technique to break a data matrix into two parts such that
D ) RC
(5)
In the current application, each column of the data matrix contains absorbances at a specific time delay after the laser flash. The rows of the data matrix are characterized by the monitoring wavelengths of the kinetic traces. The goal is to factor the data matrix so that
Rji ) i(λj)
(6)
where i(λj) is defined, as above, as the molar absorbance of the ith species at the jth wavelength and
Cik ) lci(tk)
(7)
where ci(tk) is the concentration of the ith species at the kth time point. In this application, an element Djk of the data matrix corresponds to the solution’s absorbance at the kth time point of the jth kinetic trace (or jth monitoring wavelength). Each row of the C matrix should be proportional to the timedependent concentration of one of the transients. The spectral resolution technique described above is related to the factor analysis in that eq 1 would give one of the matrix elements in the jth row of D. The precise element (which column) in the jth row would depend on the time delay under consideration. At the center of the mathematical procedure of factor analysis is a singular value decomposition (SVD)34 of the D matrix with
D ) USV ˜
(8)
where S is a diagonal matrix and U and V are unitary. V ˜ is the transpose of V. The SVD analysis was carried out with MATLAB 4.2c on a Macintosh Quadra. The square matrix ξ ˜ D, ) S˜ S, whose diagonal elements ξj are eigenvalues of Z ) D plays a central role in factor analysis. Z is called the covariance matrix. The eigenvectors of Z are given by the columns of V )C ˜ . It can be seen that the other matrix in the factored form of eq 5 is then R ) DV. The number of real components in the transient spectra can be estimated by considering the real error (RE) in the data matrix
[ ] c
∑
RE )
1/2
ξj
j)n+1
r(c - n)
(9)
computed assuming a given number n of real factors and comparing this to the estimated errors from the experiments themselves. In eq 9, c is the number of columns in the data matrix and r is the number of its rows. In the specific data
being compared to the previous spectral resolutions, c ) 16 corresponds to the number of time points or to the possible number of factors and r ) 37 is the number of wavelengths (or kinetic traces) used. The experimental errors were estimated from the noise in the kinetic traces preceding the laser pulse.35 For the current kinetic traces, the standard deviation of the noise in these absorbance differences was about 3 × 10-4. When it was assumed that there was only a single real factor in the transient spectra, then n ) 1 and RE ) 1.2 × 10-3, namely much too large to correspond to the estimated experimental error. However, assuming n ) 2 gave RE ) 4.0 × 10-4, which is of the same magnitude as the estimated experimental error. Assuming n ) 3 gave RE ) 3.8 × 10-4. Including additional factors did not lower RE significantly. For example, even with n ) 6, RE was 3.1 × 10-4. These calculations are consistent with there being at least two transient species as factors in the experimental transient spectra. Assuming those species are CB•- and (S∴N)+, an estimate for the time variation in their concentrations can also be made using factor analysis. The matrix C contains the information on the transients’ concentrations. However, both the C matrix and the R matrix generally come out of the SVD initially rotated from real concentrations and real extinction coefficients, respectively. To find the real concentrations from the C matrix, it must be rotated, and the rows associated with the errors in the data matrix have to be removed. The transformation corresponding to the rotation of coordinates is found from the truncated eigenvalue matrix ξ* and the truncated R*. The truncated matrix ξ* contains that part of the ξ matrix associated with the n largest eigenvalues, and R* contains only the first n columns of the R matrix. An analogous truncated C* matrix, which contains only the first n rows of C, is also needed. The value of n has been found above by a comparison of estimated experimental errors with the real error RE in the data matrix computed by eq 9. The transformation matrix23
ˆ T ) ξ-1R ˜ *R
(10)
transforms the C* into the final concentration matrix by
C h ) T-1C*
(11)
where C h contains the real concentrations of the n transients, with each row of C h displaying the time dependence of each transient. In eq 10, R ˆ is a matrix whose first n columns are made up of the extinction coefficients of the suspected transients. All of the truncated matrices can be formed from the results of the SVD and from a prior determination of n, the number of principal factors, which comes from the error analysis. The R ˆ matrix is constructed from previous determinations of the molar absorbances of the possible transients. C h is the final result that is being sought. In the test case, the pH 10.06 data, taken on the 3 µs time scale, was analyzed using both the spectral resolution at each time point and by the factor analysis. The same kinetic traces were used in the two analyses. For this set of data, D was a 37 (wavelengths) × 16 (time points) and C and R had dimensions of 16 × 16 and 37 × 16, respectively. R ˆ and R*were 37 × 2 matrices since n was taken to be 2 based on the error analysis above. C h and C* were 2 × 16 matrices. With the second column of R ˆ loaded with the molar absorbances of (S∴N)+, the second row of the C h matrix gave the time dependence of [(S∴N)+]. This concentration profile was fit to a twoexponential decay function, resulting in a τSN+ of 510 ns compared to 600 ns from Table 1 for the same data analyzed with the spectral resolution, time-point-by-time-point. The ratio
14920 J. Phys. Chem., Vol. 100, No. 36, 1996
Hug et al.
of the fast to slow preexponential factors was 0.92 for the [(S∴N)+] decay curve from the factor analysis compared to about unity for the same ratio from the decay curve generated from the spectral resolutions. From both analyses, the lifetimes from the slow components were much longer than the observation time, and thus, the comparison is not reliable. The agreement of the decay curves from the time-point-by-timepoint spectral resolution method in comparison to the factor analysis shows precisely how well the methods work on the data examined. Acknowledgment. The work described herein was supported by the Committee of Scientific Research, Poland (Grant No. 2 P303 049 06), and the Office of Basic Energy Sciences of the U.S. Department of Energy. This paper is Document No. NDRL3905 from the Notre Dame Radiation Laboratory. The authors thank Dr. B. Brunschwig for suggesting the application of SVD to this problem, Professor J. Wirz and Dr. R. Bonneau for the preprint of their paper, and one of the reviewers for his suggestion of the form of the unknown radical in equilibrium with the (S∴N)+ radical cation.
[SN+]+ +
[SN ]0
≈
k1k2 (k3 - k2)
2
[OH-] + k12k2
|k3 - k2|5 [SN+]+
[SN ]0
≈1-
(k32 + k3k2 + k22)[OH-]2 (A9)
k1k2
[OH-] (k3 - k2)2 k12k2 |k3 - k2|
(k32 + k3k2 + k22)[OH-]2 (A10)
5
k1k2 k12k2k3 [OH-]2 [OH-] + |k3 - k2| |k - k |3
(A11)
k1k3 k12k2k3 [OH-]2 [OH ] -φ- ≈ k3 + |k3 - k2| |k - k |3
(A7)
-φ+ ≈ k2 -
3
2
and
Appendix The exact solution to the kinetics of Scheme 1 is
[SN+] ) [SN+]+ exp(φ+t) + [SN+]- exp(φ-t) (A1) where
3
2
References and Notes
[SN+]( ) -
[SN]0 (k + k1[OH-] - k2 - ω1/2) (A2) 1/2 3 2ω
and
φ( ) 1/2(-k3 - k1[OH-] - k2 ( ω1/2)
(A3)
ω ≡ (k3 + k1[OH-] + k2)2 - 4k2k3
(A4)
with
When k2 > k3, expanding the preexponentials and the exponents in power series in [OH-] gives
[SN+]+ +
[SN ]0
≈1-
k1k2 (k3 - k2)
2
[OH-] +
k12k2
(k32 + k3k2 + k22)[OH-]2 (A5) |k3 - k2|5 [SN+]+
When k3 > k2, expanding the preexponentials and the exponents in power series in [OH-] gives
[SN ]0
≈
k1k2 (k3 - k2)
[OH-] -
2
k12k2
(k32 + k3k2 + k22)[OH-]2 (A6) |k3 - k2|5 k1k3 k12k2k3 [OH-]2 [OH-] + |k3 - k2| |k - k |3
(A7)
k1k2 k12k2k3 [OH-]2 -φ- ≈ k2 + [OH ] 3 |k3 - k2| |k - k |
(A8)
-φ+ ≈ k3 -
3
2
and
3
2
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