J . Phys. Chem. 1992, 96, 314-326
314
sumed through reaction with the various possible oxidizing radicals the formation of sulfate is no longer hampered. Further quantifications of absolute and relative yields of perthiyl and particularly sulfate still await more detailed studies on the chemical properties of the various individual intermediates and the associated reaction kinetics.
Conclusion Redox processes of di- and trisulfides induced by both photochemical and radiation chemical methods have been shown to be convenient means to generate and study perthiyl radicals. Further information was obtained on trisulfide radical anions and cations. The absolute and relative radical yields generated from trisulfides were found to depend largely on the structure of the polysulfide as shown, in particular, for the cysteine- and penicillamine-substituted compounds, CySSSCy and PenSSSPen. The most interesting finding is that perthiyl radicals in their reaction with molecular oxygen yield inorganic sulfate which, from a biochemical point of view, would constitute a pathway for nonenzymatic sulfate generation. Otherwise perthiyl radicals exhibit qualitatively similar properties as thiyl radicals, e.g., as oxidants or O2scavengers. The
respective reactivities appear to be generally lower though than those for the corresponding thiyl radicals. Oxidizing properties must also be attributed to the oxygenated radicals generated as the result of oxygen addition to the perthiyl radicals en route to the sulfate. All these intermediates, if formed in biological systems, would therefore have to be considered as potentially harmful since they may successfully degrade protecting antioxidants. Another interesting application of our findings may relate to the understanding of corrosion processes of semiconducting inorganic disulfides such as FeS2, etc. Oxidation of the S22units in these materials, e.g., via photolytically generated holes, to perthiyl-type 3-S' intermediates would provide a most plausible route to the well-known73 corrosion product sulfate.
Acknowledgment. The financial support provided by the Association for International Cancer Research (AICR) is gratefully acknowledged. (73) Bahnemann, D. In Sulphur-Centered Reactive Intermediates in Eds.; NATO-AS1 Chemistry and Biology; Chatgilialoglu, C., Asmus, K.-D., Series A, Life Sciences, Vol. 197; Plenum Press: New York, 1990; p 103.
Non a Priori Analysis of Fluorescence Decay Surfaces of Excited-State Processes. 3. Intermolecular Excimer Formation of Pyrene Quenched by Iodomethane Ronn Andriessen,+ Marcel Ameloot,* Nod Boens,**+and Frans C. De Scbryvert Department of Chemistry, Katholieke Uniuersiteit Leuven, 8-3001 Heverlee- Leuuen, Belgium, and Limburgs Uniuersitair Centrum, B-3590 Diepenbeek. Belgium (Received: May 13, 1991)
The global compartmental approach to analyze the kinetics of excited-state processes is tested on simulated and real fluorescence data of the excimer formation of pyrene quenched by CH31 in cyclohexane. An identifiability study is presented to investigate the uniqueness of the set of estimated rate constants. It is shown that all the rate constants can be uniquely determined from the relaxation times in the fluorescencedecays of four well-chosen sample preparations. The results of the identifiability study are evaluated by applying the global compartmental analysis on fluorescence decay data of the excimer formation of pyrene quenched by iodomethane in cyclohexane. The following rate constants are obtained at room temperature: kol = 2.27 X IO6 s-l, kzl = 6.7 X IO9 s-I M-I, k,, = 110 X lo6 s-I M-I, kO2= 17.1 X lo6 s-l, kI2 = 3.42 X IO6 s-I, and k,, = 27.9 X IO6 s-' M-I. The calculation of the species associated emission spectra (SAEMS) of the monomer and the excimer demonstrates that the unsubstituted intermolecular pyrene excimer in cyclohexane is emitting in the monomer emission band at 375 nm. The absorbance of the monomer normalized to the total absorbance deviates from unity at pyrene concentrations higher than 2 X 10-4 M, indicating the existence of ground-state aggregates between the pyrene molecules. The molar extinction coefficient for the dimer at 325 nm is about 2300 cm-' M-I. The value for the equilibrium constant in the ground state between monomer and dimer is found to be 2450 M-I, The existence of ground-state aggregates is further demonstrated by the change of the ratio of fluorescence intensities at the monomer and excimer emission bands as a function of excitation wavelength.
I. Introduction The global compartmental analysis of fluorescence decay surfaces'-4 has proven its power and advantages in the determination of excited-state kinetics from fluorescence decay surfaces. In this terminology each excited-state species corresponds to a compartment. Each compartment can be filled by excitation of the corresponding ground-state species or by exchange from another compartment. The compartments can be depleted by returning to the corresponding ground-state species or by transfer to another compartment. This type of analysis allows the estimation of the rate constants and the species associated emission and excitation spectra (SAEMS and SAEXS). For a bicomTo whom correspondence should be addressed. Katholieke Universiteit Leuven. 8 Limburgs Universitair Centrum.
+
partmental system the molar extinction coefficients of the ground-state species and the corresponding equilibrium constant can be determined provided the absorbance and total concentration are known.* In the identifiability study of bicompartmental systems it was shown that two physically acceptable possible solutions may exist for the rate constant^.^*^ In that case additional information is needed to select the correct set of rate constants. ( I ) Beechem, J. M.; Ameloot, M.;Brand, L. Chem. Phys. Lett. 1985, 120, 466. (2) Ameloot, M.; Boens, N.; Andriessen, R.; Van den Bergh, V.; De Schryver, F. C. J . Phys. Chem. 1991, 95, 2041. (3) Andriessen, R.; Boens, N.; Ameloot, M.; De Schryver, F. C. J . Phys. Chem. 1991, 95, 2047. (4) Ameloot, M.; Boens, N.; Andriessen, R.; Van den Bergh, V.; De Schryver, F. C. Methods of Enzymology; Academic: New York, in press. ( 5 ) Ameloot, M.; Beechem, J. M.; Brand, L. Chem. Phys. Lett. 1986,129, 211.
0022-365419212096-3 14$03.00/0 0 1992 American Chemical Society
Decay Surfaces of Excited-State Processes
The Journal of Physical Chemistry, Vol. 96, No. I , 1992 315
Lofroth6 proposed specific algorithms for two- and some three-state excited-state processes. The addition of an external quencher to the system was applied to obtain higher accuracy and precision in the estimation of the rate constants, the decay associated spectra, and the species associated spectra. Knutson et al.' introduced the use of quenched decay associated spectra (Q-DAS) to resolve the individual tryptophan emission spectra of horse liver alcohol dehydrogenase. The implementation of quenching in a global compartmental analysis program has been mentioned by Beechem et a1.* In the present paper, the bicompartmental analysis described before2q3is extended with the fitting parameters k ~and , k ~ 2the , rate constants for external quenching by a quencher Q with concentration [Q]. An extensive identifiability study is performed for a concentration-dependent two-state excited-state process with added quencher. The identifiability equations are verified on simulated fluorescence decay data. The experimental system that has been used is the excimer formation of pyrene in cyclohexane with added iodomethane. This system has been studied by M a r t i n h ~ . ~In that study, several assumptions were required to obtain all the rate constants from both steady-state or time-resolved fluorescence measurements. In this paper the only assumption made is for the rate constants to be time-independent. It will be shown that only time-resolved fluorescence measurements are required to determine all rate parameters of interest. 11. Theory 1. Matrix Solution of the Rate Equations. Consider a system (see Scheme I) with an equilibrium between two species 1 and 2 in the ground state which form upon excitation the excited-state species 1 * and 2*, respectively. These compartments can depopulate via fluorescence emission (kF), internal conversion (k,+ and intersystem crossing (klsc). The sum of these intrinsic deactivation rate constants are represented by kol for 1* and ko2 for 2*. By adding an external quencher, Q, to the system, the depletion of the excited states is enhanced by kQl[Q] and kQ2[Q] for I * and 2*, respectively. It is assumed that the quencher Q has only an effect on the excited species and does not affect the ground-state equilibrium. The exchange process from compartment I * to the compartment 2* is labeled with the rate constant k21,while the reverse process is labeled by k 1 2 . Suppose the system shown in Scheme I is excited with a Dirac delta function pulse at time zero. The rate equations for the two excited species are given by
=
[ClBII
+ c28211 exp(y1r) + [ClBl2 + c2822l exp(y2t)
(12)
with Ci
= kFiLAc,,pi(hEm) dXCm
(13)
where pi is the spectral emission density of species i, normalized to the complete emission band. Ah" is the emission wavelength interval in which the fluorescence is monitored. Equation 12 can be rewritten as flhcm,Xcx,r)= aI exp(ylr)
+ a2exp(y2t)
(14)
Alternatively,f(Xem,XCX,t)can be expressed by a matrix equation in the following way. Equations 1 and 2 can be written as At time zero the concentrations of species 1 * and 2* are denoted by [ 1 *]r.Oand [2*],=0,respectively. The time dependence of the concentration of I * and 2* is given by
with r
For the exponential factors, y, and y2, we have The matrix A can be expressed in terms of its eigenvalues y i and associated eigenvectors Ui,
A = uru-1 ( 6 ) (a) Lofroth, J.-E. J . Phys. Chem. 1986, 90, 1160. (b) Lofroth, J.-E. Anal. Instrum. 1985, 14. 403. (7) (a) Knutson, J . R.; Baker, S. H.; Cappuccino, A. G . ; Brand, L. Photochem. Photobiol. 1983, 37, Abstr. S21. (b) Brand, L.; Knutson, J. R.; Davenport, L.; Beechem, J. M.; Dale, R. E.; Wallbridge, D. G.;Kowalczyk, A . A. In Spectroscopy and the Dynamics of Molecular Biological Systems; Baley, P. M., and Dale, R. E., Eds.; Academic Press: New York, 1985; p 259. (c) Mantulin, W . W.; Beechem, J. M. J . Cell. Biol. 1988, 107. 842a. (8) Beechem, J. M.;Gratton, E. 1. In Time-Resolued Laser Spectroscopy in Biochemistry; Lakowicz, J. R., Ed.; Proceedings of S.P.I.E.;SPIE: Bellingham, WA, 1988; Vol. 909, p 70. (9) Martinho, J . M . G. J . Phys. Chem. 1989, 93, 6687.
whcrc u is the matrix with column given by
(16)
vectors u,. ~h~ matrix r is
(17)
The fluorescence
function can then be written as
f(Xem,hcx,t)= C(XCm)UerfU-~B(XCX)
(18)
316 The Journal of Physical Chemistry, Vol. 96, No. 1, 1992
Andriessen et al.
with
C represents the row vector with elements cI and c2. B is the column vector of the initial conditions and is given by c
.
c
When the elements of the vector B and C are normalized as follows: bi = b i / ( b l 62) and Zi = ci/(cI c2), eq 18 can then be written as
+
f(Aem,Aex,t)
+
= &(Aem)UeW-' B( ,ex)
(21)
with K a proportionjdity constant. The estimated B(Aex)allows the calculation of-the ratio [ 1*],=0/[2*],=~The estimated e(Aem) together with B(AeX)can be used to calculate the species associated emission spectra (SAEMS) of each of the two emitting species.23 The steady-state fluorescence spectrum F,can be decomposed into the underlying spectra of the two emitting species. The SAEMS for species i is given by
a For every set of experimental conditions, four theoretical solutions for the sets of rate constants will result.
With the global compartmental analysis program the rate constants kol, k 2 ] ,k ~ ,kO2, , k I 2 ,k ~and ~ 6,, and PI can be determined at once from a suitable chosen fluorescence decay surface. In the next section we investigate the appropriate set of experimental decay curves for the determination of these parameters. 2. Identifiability Study. An identifiability study investigates the uniqueness of the solution recovered from a given set of error-free data. For the case considered in this paper, the fluorescence delta-response function is known to be biexponential (eq 14). The approach then is to construct analytical expressions using yI, a2,and y2 from which the rate constants, 6 , and Z, have to be obtained. The six rate constants kol, k 2 , , kQl, kO2.k 1 2 and , k ~ (Scheme , I ) have to be determined from the relaxation times of the fluorescence decays obtained for the various sample preparations. For each combination of concentrations for M and Q given by CY,?
To allow the determination of the six rate constants, at least three different combinations of ( [MIk, [Q]J are required. Several acceptable solutions for the rate constants may be obtained from uIAiand b 2 k / . This will be investigated blow. For each solution for the rate constants, the unknowns b, and T I have to be determined from the Markov parameters mP2s5For each recovered fluorescence delta response, the Markov parameters m,are given by mik/ = f f l k f f i l k / + l Y 2 k / Y ' 2 k /
(25) where A' denotes the ith power of matrix A. As it is assumed that ihe quencher Q has only an effect on the excited-state species, b , will be independent of the quencher concentration, [Q]. = &A'B
WkJlk+lJ+l)
E
-Sk/Qk+l./+~+ S~+I./+IQM (37)
Note that T(k,l)k+l,l+I), U(k,llk+l,l+l), V(k,llk+l,l+l), and W(k,l)k+1,I+ 1 ) are also functions of the excitation and emission wavelength. Equations 23 and 24 are the basic equations for the determination of the rate constants. Depending on the chosen set of sample preparations ([MIk, [Q]!), these equations have to be combined in order to get expressions for the separate rate constants. Once the rate constants ace known, eq 33 is the basic equation for the determination of b, and SI. Only experiments
The Journal of Physical Chemistry, Vol. 96, No. 1 , 1992 317
Decay Surfaces of Excited-State Processes with a variation of the emission wavelength XCm, i.e. a change of i;,, will be discussed below. For experiments in which the excitation wavelength, Xcx, is vaned, a change in the 6l.k value results. Similar procedures to the ones exposed in the following sections can be used in this case. The determination of the rate constants, 6, and PI from various sets of sample preparations ([MIk, [Q]J will now be discussed in a systematic way. Case 1. Three Different Combinations of ([Mlk, [Q)). Several realizations are represented in Scheme 11. I . A . The Realization ( [ M I , ,[Qll), ( [ M I 2 , [QIJ and ([W3, [QI3) (Scheme I I A ) . From u I I l , and 413) (eq 23) the following combinations of the rate constants can be determined: A 1 kol + ko2 kl2
For the other combination {([MI,, [Qll);([MI3, [Q]j)l, an expression similar to eq 42 is obtained. Thus, collecting decay traces ct three emission wavelengths yields twice a unique solution for 61.1, resulting in a more accurate parameter recovery than in the case where o$y two emission wavelengths are used. The knowledge of b,.] allows the determination of b1,2,b,.,, and f,. I.B. The Realization ([MI,[Qll), ( [ M I , [ Q l A and ( [ g 2 , [QI1)(Scheme IIB). From ulkl(eq 23) the following equations are obtained
+
(47) From
fJ2kl
(eq 24) it can be shown that
From u I I Iand u Z l lwe have k12 = -7212D1
+ ~ [ M I+I
QIII
[(C[MI,+ Q I I I + )~ 4(D,C[Ml lQ211)11'21(49)
and
E1 1 k o i + ~ Q I [ Q I=I - ~ I ~ - D I - C [ M I I - Q ~(50) ~~ Two mathematical solutions for k12and E, have to be considered. k,,, k,?, and ko2can be written in terms of kol: ~ Q =I
Using eqs 38-40 in eq 24 leads to a fourth-order equation in k ~ , . If only one mathematical solution exists with positive values for all rate constants, the system is identifiable as far as the rate constants are concerned. If more than one positive solutip for the rate constants exist, the nonnegativity requirement of 6, and i;, may yield a unique solution. For each acceptable set of rate constants, the values for 6, and PI can be recovered as follows: Considering eq 33 at two emission wavelengths Xiem and "A; for the combin-ation {([MI,, [QI1);([M2, [QJ2))yields-after elimination of 61,Z-a quadratic equation in bl,, given by
(E1 - koi)/[QIi
k02 = DI - B[QIi
(51)
+ E , - ko1
(53)
These expressions (eqs 51-53) are used in u212(eq 24), resulting in a quadratic expression in kol for each value of k12 FkOl2+ Gkol + H = 0
(54)
F = -(I - [Qlz/[Q11)~
(55)
with
b21,1(T,Uj- TjUj) + Bl,,(UjW, + TjV, - TjV, - UjW,) = VW,- VjWj (41) T, U,V , and Ware defined by eqs 34-37. The subscripts refer to the corresponding _emission- wavelengths. Two solutions for 6l,l and 6,,2 are obtained. Equation 33 at two emission wavelengths Am: and Xjm for the combination I( [MI ,, [Q] ,);([MI,, [Q],)) yi_elds-after elimination of b,,3-a similar quadratic equation in 6,,], When the two quadratic equations in 6,,,are linearly independent, two emision wavelengths_willsuflice to determine a unique solution for 6,,]. In this case 61.2 and 61.3 are also known and the Z;, values can now readily be calculated from eq 28. If the two quadratic equations (eq 41) for the two combinations of ([MIk, [Q]J are linearly dependent, a third emission wavelength has to be included. Two eqs 41 at three emission wavelengths AIcm, XZCm,and for the combinati_on {([MII. [Q] ]);([MI2, [QI2))yield the following expression for 6]*,
where XI,, Y,, and Z, are defined at emission wavelengths XIem and ,A; X, = T,Ul - T,U, VI, = UJW1
+ T,V, - T,VJ- UIW,
z, = VIWJ - V,WI
(43) (44) (45)
H = IC[Mli + Ei[Qlz/[QliI{~i+ B([Q12 - [Qli) + Ei(1 [Qlz/[QIi) + k12J - ~ I z C [ M I I~ 2 1 2(57) Four different sets of kol, k2,, k,,, ko2,k12,and k ~ are 2 obtained. Only the sets which contain only nonnegative values for the rate constants can be accepted. For each acceptable set of rate constants, the values for 6, and T I can be recovered as follows. The set {([MI,,[Q],);([M],, [QI2)]colkcted at one emission wavelength yields a quadratic equation in 6]*,(eq 33). If for the considered set of rate parame_tersone of the two solutions for 61.1 is negative, the parameters b , , , and Z;, can be determined with decay data collected at a single emission wavelength for each acceptable set for the rate constants. If both solutions are positive, the decay traces collected at two emissio! wavelengths, and ,A,; yield the following expression for 61.1
6,l =
u,WI - UIwj + V,)uJ - (TJ +
(58) c)ul
resulting in a single solution for JI,]for the considered set of rate constants. The set I( [MI [Q] ,);( [ MI2, [Q] ,)] at two $mission wavelengths X,Cm and yields a quadratic equation in b , , , ,similar to eq 41. resulting in two solutions. One of the two will correspond to the
318
The Journal of Physical Chemistry, Vol. 96, No. 1. 1992
SCHEME III: Schematic Representation of the Sample Preparations Used in the Case Where Four Combmtions of ([MIb [Q],)Are Uaed"
Andriessen et al. 11.B. The Realization ([MI,[QIA ([MI,[QM, ([MI,[QIA and ([MI2, [QI1) (Scheme IIIB). This is an extension of the situation depicted in Scheme 11s or I1C (case 1B or IC). Equation 24 can be written as follows: 6211
3 or 4 theoretical solutions
= 11+ JilQ1, + K[Ql?
with I1 = kOlkO2 + kOIkl2 + ko2k2,[Ml, JI
(koi
+ k21[Mll)kQ2 + (ko2 + kl2)kQl K E k~lk,,
2 theoretical solutions
1 theoretical solution
value calculated according to eq 58. Thts, using two emission wavelengths yields a unique solution for b , , , for the considered set of rate constants, from which the remaining parameters, b1,2 and the two i;, values can be determined using eqs 26-32. When the decays of ([MI,, [Q],) and ([MI,, [Q],) are-measwd at three emission wavelengths, a linear equation in b l , l or bl,2can be obtained. I.C. The Realization ( [ M I , ,[QII), ([MI,,[QlA and([MI2, [ e l 3 ) (Scheme IIC). An analogous procedure for the deterpication of the rate constants can be followed as in case IB. b,,,, b1.2,and f, values can be determined using with the same procedure as was exposed in case IB. 1.D. The Realization ([MI,, [QII), ([MI,, [QIl), and ( [ M I 3 . [Q12)(Scheme [ID). An analogous procedure as in case IA can be followed, resulting in four sets of rate constants. For_each acceptable set of rate constants, a single solution for the b, and Z, values can possibly be obtained from the decay data collected at two emission wavelengths. A single solution is guaranteed from the decay data collected at three emission wavelengths. I n all cases (case IA-D), four sets of rate constants can be obtained from eqs 23 and 24. The existence of only one set of positive rate constants will result in the unique solution of the system. If more than one set of Fsitive rate constants exists, the requirement of nonnegativity for b, and E , may lead to the unique solution. If not, the addition of the fluorescence decay data (eqs 23 and 24) of other combinations of ([MIk, [Q],) is necessary to uniquely determine the system (see below). Case 11. Four Different Combinations of ([MIk, [Q],). II.A. The Realization ([MI,, [QII), ([MIz,[Q12), ( [ M 3 , [QIA and ([Mj4, [QI4) (Scheme IIIA). This realization can be considered as a combination of the realization ([MI,, [Qll), ([MI*. [QIJ, ([M13, [QIJ and the realization ([MI,, [QIA ([MIt, [Qld, (IM14. [QI4), which is an extension of the situation depicted in Scheme IIA. Equations 38-40 can be obtained from eq 23. The two fourth-degree equations in k,, can be combined into a third-degree equation, if the two equations are independent. This results in three mathematical solutions for_the rate constants. For the determination of the b, and i., values, the same procedure as for case IA or ID can be followed. When the fluorescence decay of each of the sample preparations is measured a t the same two emission wavelengths, three quadratic equations in b l , l (eq 41) can be constructed for each acceptable set of the rate consta_nts. For at least one set of rate constants a single solution for b , , , will be obtained.
(60) (61) (62)
For three different values of [Q], at the same value [MI, we can dctermine eqs 60-62. From eqs 46 and 62 we get solutions for kQ1 and k ~ 2 .However, one does not know which value has to be assigned to k,,, respectively to kQ2. For each pair of solutions for k,, and kQ2, the following procedures gives the corresponding values for the other rate constants. Defining L
"The number of theoretical solutions for the rate constants are indicated.
(59)
M
kol k02
+ k12
(63) (64)
eq 38 can be written as A = L + M (65) This value and the rate constant k 2 , can be determined from the g I k lvalues. From eq 65 and
(66) J', I J, - ~ Q & ~ I [ M=] kI ~ 2 L+ kQIM values are obtained for L and hf if k,, # k ~ 2 .Equation 60 yields I, - LM k02 = (67) k2, [MI I so that kl2
= M - ko2
(68)
This yields two sets of rate constants. If in only one set all rate constants are nonnegative, the rate constants are uniquely defined. If not, the unique solution can possibly be obtained by rejecting the solutions yielding negative vslues for b , and 2,. For the determination of the b , and El values, the same proccdure as for case IB or IC can be followed. Decay curves collected at a-single emission wavelength will yield a quadratic equation in bI,,for the combination I([M],,[Q1,);([Mll,[Q]2)} and a quadratic equation in b , , , for the combination (([M]l,[Q],);([MI 1,[Q]3)).Combin_ationof the two quadratic equations yields a single solution for bl.,, provided that the equations are independent. Otherwise, the decay data collected at two emission wavelengths are needed to yield a single solution for b , , , for the considered set of rate constants. 1I.C. The Realization ([MI,,[QII),([MI,[QM. ([M2. [QIIL and ([MI2, [Q12) (Scheme IIIC). This is an extension of the situation depicted in Scheme IIB. For three different combinations of ([MI,, [Q],) A (eq 38) can be determined. B and C are given by cqs 46 and 47, respectively. Two different concentrations of M, [MIl, and [MI,, at the same concentration of Q, [QIl,yield u ~values ~ , (eq 24) which are linear in [MI. The slope PI and the intcrcept N , are NI = (koi
+ k~i[Qli)(k02+ ~ Q Z [ Q ]+I )( ~ O I k~i[Qli)k12 (69)
PI = (ko2 Equations 70 and 47 yield
4-
k~~[Qli)k21
ko2 + k~z[Qli= Pi/C
(70) (71)
k,, and kQz can be determined from eq 71 at two different con-
Decay Surfaces of Excited-State Processes
The Journal of Physical Chemistry, Vol. 96, No. I, 1992 319
different concentrations of M and Q have to be included and (2) kQl must not equal kQ2. The use of three different combinations ([MIk,[Q],) may result Ri = Ni - ~ O & Q I [ Q ] I - ~ Q I ~ Q ~ [ Q ] I ' in four physically acceptable sets of rate constants. If there is more than one set in which all the rate-constants are nonnegative, = koi(ko2 + k12) (kOikQ2 + ki.$~i)[Q]i (72) thc requirement of nonnegativity for b, and P I or the addition of more combinations ([MIA,[Q],) may lead to the unique solution A similar equation, R2, can be written for [QI2. One obtains for thc rate constants. (73) R I - R2 = (koik~2+ ~ I Z ~ Q I ) ( -[ Q[QIJ II The use of four combinations ([MI,, [Q],) results in the unique solution for the rate constants if the combinations are chosen as Equation 38 together with eq 73 yields kol and k12,provided that depicted in Scheme I1lC. Other realizations for ([MIk, [Q],) can kg1 # kQ2. Thus, the experimental setup depicted in Scheme lTlc result in two sets of rate constants (Scheme IIIB) or three sets will always yield a unique solution for the rate constants, provided of r?te constants (Scheme 1IlA). The condition of nonnegativity that k ~ and , k ~ are 2 different. for b, and F, or the addition of more sets of ([MI,, [Q],) may yield For the determination of the b, and Zl values, the same prothc unique solution for the rate constants. cedure as for case 1B or IC can be followed. Decay curves colFive sets of ([MIk, [Q],), satisfying condition 1, lead to the lected at a _single emission wavelength will yield a quadratic unique solution for the rate constants, on the condition that the equation in 61.1 for the combination {([M]l,[Q]l);([M]l,[Q]2)) and determinant of the matrix of eq 74 is different from zero. a quadratic equation in bl,2for the combination {([M],,[Q],);The b , and F , values can be determined from decay data ([MI2,[Ql2)).Combination of the two s_olutionsfor both quadratic collected at one or two emission wavelengths if at least one comcquations yields a unique solution for b,,,and b,,, provided that bination [([MIk,[Q],); ([MIk, [Q]r+l)}has been used. If [MI is the solutions are not both the same. Otherwise, decay data different for all ( [MIk,[Q],), the decay data collected at two or collected at two emission wavelengths are needed to yield a unique three emission wavelengths are necessary to determine the b , and solution for the system. F, values. Case 111. Five Different Combinations of ([MI,, [Q],). In this As was reported earlier?*5a system like Scheme I without added case, eqs 38-40 or eqs 38,46, and 47 can always be constructed. quencher can sometimes have two physically acceptable solutions Equation 24 can be written as follows: for the rate constants. The only way to determine the unique solution without adding quencher is to try to get information about 6 2 k i = WQI? + YQ1,+ x[MI,[Ql/ + VMlk + (74) one or more parameters, e.g., one or some rate constants and/or b, and/or PI. If this procedure is not possible, one can determine with the unique solution for the rate constants by adding quencher to = kQlkQ2 (75) the system. According to the experimental setup depicted in Scheme l l l c or by using five or more combinations ([MI,, [Q]/), w = (ko2 k 1 2 ) k ~+i koik~2 (76) thc unique solution for the rate constants can be determined.
centrations of Q, [QI1,and [QI2. Once kQ2is known, eq 46 gives kQl. Define
-
= k21kQ2
(77)
Ill. Experimental Methods
y = kO2k21
(78)
The synthetic data were generated as described bef0re.j The data analysis program described in our earlier work3 was extended to allow for the fitting parameters kQ, and kQ2. Fluorescence decay curves were obtained using the 325-nm excitation of a Spectra-Physics synchronously pumped, modelocked, cavity-dumped, frequency-doubled DCM dye laser with single-photon timing detection. All fluorescence decay curves were collected in 1 / 2 K data points of the multichannel analyzer and contained between 5 X IO3 and IO4 counts at the peak. Details of the fluorescence lifetime apparatus and the associated optical and electronic components are described elsewhere.I0 The fluorescence decay curves of the excimer forming solutions were collected in the front-face configuration with I-mm quartz cells. The emission polarizer was set in the magic angle configuration. Fully corrected steady-state fluorescence spectra of the excimer forming solutions were recorded in the front-phase configuration on a SPEX Fluorolog 21 2 at 334 nm excitation. The fluorescence intensities of the M solutions of pyrene were corrected for the absorption of iodomethane by multiplying the intensity by I@ICHl']d12, with 4334 nm) = 1.5 M-I cm-l the molar extinction coefficient of iodomethane at 334 nm and d = 1 cm, the path length of the cell. Pyrene (zone refined), cyclohexane (Merck; Uvasol), and iodomethane (CHJ) (Aldrich; Gold Label) were used without further purification. Solutions of pyrene with the quencher were prepared in the dark to avoid photodegradation. The solutions were degassed using five freeze-pump-thaw cycles. All measurements were done at room temperature.
= kOl(kO2 + kl2)
(79)
Considering eq 74 at the five different combinations of ([MIk, [Q]J results in a system of equations from which the following combinations of the rate constants can be determined if the equations are independent: kQ2can be determined from eqs 40 or 47 and 77, kQl from eqs 40 or 43,77, and 75, kO2from eqs 40 or 43 and 78. The expression for k I 2is then given by
kI2 =
- AkQ2
- k02(kQl - kQ2) (kQI - kQ2)
(80)
provided that kQ, # kQ2. kol can be calculated from eq 79. There are two conditions to find a unique mathematical solution for the rate constants starting from the fluorescence decay data of a combination of five different sample preparations ([MIk, [Q],). kQl must not equal kQ2and the determinant of the system according to eq 74 must not be zero; otherwise the system cannot lead to a unique solution. At least two different values for [MI and [Q] are required. The number of different emission wavglengths, needed for the determination of a unique solution for the b, and f l values, depends on the combinations of ([MI,, [Q]/). If [Q] is changed while [MI, is constant, a single emission-wavelength can be sufficient to yield the unique solution for the b , and Z, parameters. Two emission wavelengths always yield the unique solution. If [MI is changed while [Q], is constant, two emission wavelengths can be sufficient to yield the unique solution while three emission wavelengths always yield the unique solution. The conclusions of this identifiability study can be summarized as follows: to determine the rate constants two conditions have to be fulfilled, independently of the number of different sample preparations. ( I ) For each combination ( [MI,, [Q],), at least two
IV. Results and Discussion 1. Computer-Generated Data. To investigate the performance of the new implementation of the compartmental analysis with (IO) Boens, N.: Janssens. L.;De Schryver, F. C. Biophys. Chem. 1989,33, 17.
320 The Journal of Physical Chemistry, Vol. 96, No. 1. 1992
Andriessen et al. SCHEME IV
Figure 1. Three-dimensional plot of the variation of the relaxation times 7]-' (A, top) and (B, bottom) (eq 5) as a function of pyrene [MI and iodomethane [Q] concentration. y i was calculated using the rate constant values of Table I . TABLE I: Rate Constants for hrene Photophysics with Added Iodomethane in Cyclohexane According to Birks" and Martinho' kol = 2.25 X IO6 s-I kO2= 15.5 X IO6 s-I k2]= 6.7 x 109 M-1 S-I k I 2= 6.5 X IO6 s'I k,, = 1 I O X IO6 M-' s-I k,, = 17 X IO6 M-I s-I TABLE 11: Simulated Decay Times and -yL1 (in ns) for Different Pyrene, [MI, and Iodomethane, [Q], Concentrations in Cyclohexane According to the Rate Constant Values of Table I
[Qlr [MI,, M
0 M 0.03 M 2X 322.7 158.2 44.5 43.4 5X 230.6 134.7 43.0 41.7 8X 182.7 118.9 4i.5 40.0 IXlO-) 162.1 111.1 40.4 38.9 2 X IO-) 1 1 1.8 89.2 34.9 33.1
0.05 M 0.07 M 94.6 118.3 41.9 42.6 105.9 87.6 39.9 40.9 97.1 82.4 39.0 38.0 79.8 92.6 37.8 36.7 79.3 71.8 30.7 31.9
0.1 M -
73.0 40.7 70.1 38.5 68.0 36.3 66.9 34.9 63.6 28.9
0.2 M 44.4 35.2 46.1 32.0 47.1 29.6 47.6 28.3 49.1 23.4
added quencher, fluorescence decays of pyrene at several concentrations of pyrene and idomethane in cyclohexane were simulated. The rate constants at room temperature for the pyrene system without quencher published by Birks et at." were used in combination with the rate constants of quenching by iodomethane published by Martinhog (Table I). According to Martinho, the quenching of the pyrene system by iodomethane is kinetically controlled resulting in time-independent quenching rate constants for the monomer and excimer quenching. Therefore, the kinetic scheme can be represented by Scheme I . Using the rate constants reported in Table I, the decay times -y,-' and -y2-' were calculated for the different pyrene and iodomethane concentrations using eq 5 . The calculated decay times are listed in Table I1 and plotted in Figure 1. ( I 1 ) Martinho, J . M. G.; Pereira, V. R.; Goncalves, Da Silva, A.; Conte,
J. C. J . Photochem. 1985, 30, 383.
The decay times -y;l are different for each different combination ([MI,, [Q],), so that they cannot be linked in a global bicxponential analysis. Figure 1 shows the variation of the two decay times TI-'and -y
