-
J. Phys. Chem. 1991, 95, 2041-2047
P B H - 'P*BH due to some quinone depleted RCs. However, in this case and especially at low temperatures the fluorescence yield should depend exponentially on the electric field, (P(E)/@(O) = (exp[-jiE/kT]) = O.S(kT/pE) sinh (pE/kT'),in contrast to the quadratic dependence actually observed (the brackets ( ) denote averaging over all orientations). Therefore, the contribution of delayed components of the fluorescence should be negligible as also expected for a thermally activated process at low temperatures. Furthermore, low light intensities as used in the above experiments rule out any artifact due to syperproportional representation of a minority of quinone depleted RCs due to saturation effects. Therefore, we attribute the electric field effect on the fluorescence to changes in the primary charge separation rate. Consequently, the observed anisotropy clearly shows that unistep primary charge separation according to 'P*BH P+BH- occurs. Such a unistep ET is only possible in terms of the superexchange mechanism.' Because the fluorescence quantum yield increases with decreasing quenching rate the possibility remains that a few RCs with a slow charge separation rate dominate the fluorescence signal. Whether the observed anisotropy on the fluorescence holds for the majority of RCs or for a minority characterized by slow charge separation can only be decided on the basis of picosecond time-resolved fluorescence measurements in an electric field. In the context of the recent femtosecond data2s*26 claiming a two-step mechanism at room temperature and ET calculations4 our results would be consistent with (i) a transition from the +
(48) Bixon, M.; Jortner, J.; Michel-Beyerle, M. E. Blochim. Blophys. Acra. in press.
2041
unistep superexchange mechanism at low temperature to a thermally activated sequential process at high temperatures or (ii) a slow superexchange charge separation rate k pertaining to a small subset of RCs. Comparing with femtosecond data, it should be kept in mind that electric field experiments are usually confined to matrices like polymer or Langmuir-Blodgett films. Since medium effects on the mechanism cannot be ruled out a priori, femtosecond absorption measurements on PVA samples are important. Nevertheless, independent of the origin of the low-temperature steady-state fluorescence signal, the electric field effect reported here mirrors unistep superexchange charge separation. Excitation dichroism of the electric field modulated fluorescence yield gives access to angles K which differ by 3 1 for both mechanisms. These angles and in particular the associated dichroic ratios (0.75 and 1.25, respectively) differ significantly. Thus the conclusion derived even holds if one allows for considerably larger uncertainties in the direction of dipole and transition moments, as compared to an earlier approach.28 Such uncertainties may also include possible changes in the geometry of the cofactors in the dominant subset of RCs. Very recently, the excitation dichroism of the electric field effect has been checked for consistency over a wide spectral range, Le., for five different transitions, showing that no significant geometrical changes have to be envisaged.44 Acknowledgment. We express our gratitude to Dr. H. P. Braun for introducing us to the technique of PVA film preparation, and for many helpful suggestions. Financial support from the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 143) and the Alfried Krupp von Bohlen und Halbach Stiftung (U.E.)is gratefully acknowledged.
Non a Priori Analysis of Fluorescence Decay Surfaces of Excited-State Processes. 1. Theory+ Marcel Ameloot, Limburgs Universitair Centrum, Universitaire Campus, B-3590 Diepenbeek, Belgium
N d l Boens,* Ronn Andriessen, Viviane Van den Bergh, and Frans C. De Schryver Department of Chemistry, Katholieke UniversTtez Leuven. B-3001 Heverlee, Belgium (Received: July 3, 1990; In Final Form: September 27, 1990) Fluorescence can be collected under a variety of experimental conditions. Simultaneous, or global, analysis of the constituent decay curves of a fluorescence decay surface can be performed by linking the common parameters in the various expressions used in the fitting process. Fluorescence decay surfaces can be globally analyzed in terms of the rate constants and species associated spectra by the so-called global compartmental analysis [Beechem et al. Chem. Phys. Left. 1985, 120.4661. Up to now, this method required that the decay curves be properly normalized and that the ratio of the absorbances of the specie8 in the ground state be known. However, normalization between collected decay curves is not always feasible nor is knowledge of the ratio of the ground-state absorbances always available. It is demonstrated in this paper that the conditions of proper normalization and knowledge of the ratio of the absorbances are not required for the direct analysis of two-state excited-state intermolecular processes. The required combinations of experiments necessary to achieve the direct analysis can be summarized as follows: n.-nm 1 n,, + nm, where n- and n,denote the number of different concentrations and emission wavelengths, respectively. When the total concentration of the ground-state species and the total optical density are known, the equilibrium constant of the ground-state process and the extinction coefficients of the absorbing species can be determined from timeresolved fluorescence measurements. 1. Introduction
Time-resolved fluorescence measurements allow to unravel the kinetics of excited-state processes. The decays are collected under a variety of conditions to elucidate different aspects of the process. *Towhom correqmdence should be a d d r d . 'Dedicated to Prof. A. Weller on the occasion of his birthday. A preliminary account of this work was presented at the meeting of the Society of PhoteOptical Imtrumentation Engineen, January 15-17,1990, h Angela, CA. 0022-3654/91/2095-2041$02.50/0
The excitation and/or emission wavelength, pH, concentration, temperature, etc., can be varied. The resulting data surface can be analyzed in terms of the parameters of interest, Le., the rate parameters and the emission spectra of the species, the so-called species associated emission spectra. Several approaches have been suggested to extract the required information from the fluorescence data surface. When the relaxation times are not too similar and the ratio of the preexponential factors of the exponential functions is not too large or is close to zero, the individual decay curves can be analyzed separately in terms of relaxation times and preexponential factors. Q 1991 American Chemical Society
2042 The Journal of Physical Chemistry, Vol. 95, No. 5, 1991
However, a simultaneous or global analysis'-s of related decay curves should be applied whenever possible. For example, by linking relaxation times over related decay curves a much better determination of the number of required exponentials and of the values of the fitting parameters can be The combination of the values obtained for the rela'xation times and the preexponentials can then lead to the determination of the rate constants of the excited state. In some cases, it may be necessary to assume that at some wavelength the relaxation of just one species can be followed. Once the rate constants of the excitedstate process are known, the species associated spectra can be ~ a l c u l a t e d .Once ~ a model can be proposed, the complete data surface should be analyzed directly in terms of the parameters of the model: the rate constants and the species associated spectra. This so-called global compartmental analysis allows one to link rate constants and species associated spectra Over the fitting curves for each of the individual curves of the data surface so that one can take advantage of the global a n a l y ~ i s . ~ *In~ the current implementation of this method assumptions have to be made on the relative fraction of the incident light absorbed by each species. Apart from the numerical aspect of the analysis of excited-state processes, it has to be realized that the preexponential factors and the relaxation times are not always related to a unique set of rate parameters and spectral contours.' This is translated into large uncertainties on parameter estimates and dependence of initial guesses when performing a least-squares analysis of experimental data. Consequently, it is important to know what type and number of experiments have to be performed to obtain a unique solution for the kinetic and spectral parameters in a system undergoing excited-state processes. This identifiability problem is independent of the accuracy of the measured data and is not related to the way in which the decay curves are collected, either with pulse or frequency methods, and analyzed, either individually or globally. In an identifiability study error-free data surfaces are assumed so that an analytical treatment can be developed. The resulting analytical equations also allow to obtain the rate constants and the species associated from the relaxation times and the preexponential factors. The relaxation of an excited-state process can be described by a system of coupled linear differential equations of first order. This makes excited-state systems formally equivalent with compartmental Many papers dealt with the identifiability problem in compartmental systems.&'o However, a separate study of the identifiability of excited-state processes as detected by fluorescence is appropriate. The multidimensional nature of the fluorescence phenomenon, i.e., one can use several independent ( I ) Janclrens, L. D.; Boens, N.; Ameloot, M.; De Schryver, F: C. J. Phys. Chem. 1990,94,3564. ( 2 ) (a) Knutaon, J. R.; Beechem, J. M.; Brand, L. Chem. Phys. Lett. 1983, 102,501. (b) Ameloot, M.;Beechem, J. M.; Brand, L. Biophy$. Ch" 1986, 23,155. (c) Boens, N.; Jancurens, L.D.; De Schryver, F. C. Biophys. Chem. 1989, 33, 77. (d) Mfroth, J.-E. Eur. Biophys. 1985, 13,45. (e) Weidner, R.; Gcorghiou, S. In Time-Resolved Law Spectroscopy in Biofhemistry II; Lakowicz, J. R., Ed.;Proceedings of S.P.I.E.;SPIE: Bellingham, WA, 1990; Vol. 1304, p 717. (3 Beechem, J. M.; Ameloot, M.; Brand, L. Anal. Imtrum. 1985,14,379. (41 (a) Mfroth, J.-E. Anal. Inrtrum. 1985,14,403. (b) Mfroth, J.-E. J . Phys. Chem. 191)6,90, 1160. (5) (a) Beechem,J. M.; Gratton, E. In Time-Resolved Laser Spectmmpy in Biochemistry; Lakowicz, J. R., Ed.;Proceedings of S.P.I.E.;S P I E Bellingham. WA, 1988; Vol. 909, p 70. (b) Weidner, R.; Gcorghiou, s. In Time-Resolved Laser Spectroscopy in Biochemistry; Lakowicz, J. R., Ed.; Proceeding of S.P.I.E.; SPIE Bellingham, WA, 1988; Vol. 909, p 402. (c) Becchem, J. M.;Gratton, E.; Ameloot, M.; Knutson, J. R.; Brand, L. In Fluorescence Spectroscopy, Vol. I : Principles and Techniques; Lakowicz, J. R., Ed.; Plenum Preu: New York, in pras. (6) k c h e m , J. M.; Ameloot, M.; Brand, L. Chem. Phys. Lett. 1985,120, 466.
(7) Ameloot, M.; Beechem, J. M.; Brand, L. Chem. Phys. Lett. 1986,129, 211. (8) (a) Jacqucz, J. A. Compartmental Analysls in Biology and Medicine; Elsevier: Amsterdam, 1972. (9) Galfrey, K. Compartmental Models and Their Application; Academic Preu: London, 1983. (IO) Anderson, D. H. Compartmental Modeling and Tracer Kinetics; Lecture N o t a in Biomathematics, Vol. 50; Springer-Verlag: Berlin, 1983.
Ameloot et al. experimental axes to study a given problem, makes it different from the commonly considered compartmental systems. Furthermore, a large number of data points of high quality can be obtained with time-resolved fluorescence measurements. This is usually not possible in other compartmental systems. In addition, the a priori information available in the common compartmental systems may also be different in fluorescence studies. The identifiability roblem has been treated for two-state excited-state processes.P In that report properly normalized decay curves were assumed. In this paper, the case of a two-state excited-state processes with a known concentration dependence in the forward process is discussed in terms of unnormalized decay curves. It is also shown that for the considered system the ratio of the absorbances of the species in the ground state can be determined from the fluorescence decay surface. Previously, it was assumed that the ratio of the absorbances was known a priori. When this information about the absorbances is combined with the value of the total concentration and of the optical density in the ground state, the equilibrium constant of the ground-state process can be determined together with the respective extinction coefficients of the absorbing species. This means that the equilibrium constants of both the ground- and excited-state process can be determined at once. 2. Theory 2.1, General Description of Excited-State Processes. Consider a causal, linear time-invariant system consisting of n different species or compartments in the ground state. The species may interact or not, but are assumed to be in equilibrium. The concentration of species i will be denoted by x,, i = 1, ...,n. Excitation of compartment i leads to the corresponding concentration x,*(t) in the excited state. The relaxation of the system after unit delta excitation is given by the following system of differential equations
with the initial conditions x,*(O) = b,
(2)
The coefficients k,,, k,, 1 0, represent the apparent rate constants of transfer of species j to species i. The subscript 0 denotes the ground state of the considered species. This means that is given by the sum of radiative and nonradiative deactivation rate constants. The coefficient b, is the fraction of the incident light that is absorbed by species i in the ground state and is given by, as shown in Appendix A (3) with (4)
and where c, is the decadic extinction coefficient and d is the path length of the excitation light in the measuring cuvette. Obviously, b, depends on the considered excitation wavelength and the set of concentrations of the various species in the ground state. The system of differential equations (1) can be written in matrix notation as dX*(t) = AX* dt where X*(t) = [ x , * ( t ) ] is a ( n X 1) vector; A = [au] is the (n X n ) system or transfer matrix. The matrix A indicates the con-
The Journal of Physical Chemistry, Vol. 95, No. 5, 1991 2043
Fluorescence Decay Surfaces. 1 nectivity between the compartments in the excited state. The elements of A are
aij = k,,
i #j
(6)
where s, is (-1)"' times the sum of all the m-square principal minors of A, Le., the minors corresponding with the diagonal elements of A. The characteristic polynomial of A is also given by
fi (y - 7,)= y" - ulyP1 + u2yr2 - ... + ( - 1 ) " ~ ~ (19)
i- 1
with
The solution of eq 5 is given byio X * ( t ) = exp(tA)B
(8) where the exponential of the matrix A, exp(tA), is defined by the Taylor series of the function (tN2 (tN3 +- 3! + ... l! 2!
exp(rA) = I + (tA)
+
(9)
and where B = [b,] is the ( n X 1) vector of the initial conditions. The matrix A can be expressed in terms of its eigenvalues yi and eigenvectors Ui A = uru-1 (10) where U is the matrix with columns vi; the elements of the matrix l' are given by rii = y, and rij = 0 for i # j . Equation 8 can then be rewritten as
X*(r) = U exp(tF)U-IB
(11) The fluorescence delta response, 4(t), observed within the selected emission band depends on the emission spectrum and the radiative deactivation rate of each species and is given by 4(t) = CX*(t) (12) where C is a (1 X n) vector. The elements ci are, apart from some instrumental proportionality factors, given by
where ki, is the radiative deactivation rate of species i ; p,(Xem) is the spectral emission density of species i, normalized to the complete emission band; AXm is the emission wavelength interval in which the fluorescence is monitored. Using eq 8 the scalar function # ( t ) can be written as # ( t ) = C exp(tA)B
(14)
or equivalently, by using eq 11
4(t) = CU exp(tr)U-lB (15) This means that for each combination of excitation wavelength, concentration, and emission wavelength, # ( t ) will be given by a sum of exponentially decaying functions
Due to the properties of the matrix A the eigenvalues can be shown to be negative.I0 The eigenvalues are the negative of the inverse of the relaxation times usually encountered in the fluorescence literature. The steady-state emission spectra F,in terms of A, B, and C are obtained by integration of eq 15 This leads to a general expression for the calculation of the condition for isoemissivc points, as is shown in Appendix B. 2.2. fdentifabiliry Equations. Each combination of experimental conditions such as concentration and emission wavelength yields a corresponding set [a,,y,].The elements of B, A, and C have to be determined from these sets of values. To determine these elements, the following approach has been used. The characteristic polynomial of the system matrix A is given by det (71 - A) = yn + slyp1+ s2yR2+ .., + s, (18)
"I
I
4
Ti, u2
w
yirj,
=3
,
YlYjYk,
'Jn
e Yiyp*.?'n
r
