J. Phys. Chem. 1992,96, 6331-6342 (12) Ruasse, M.-F.; Aubard, J.; Galland, B.; Adenier, A. J. Phys. Chem. 1986,90,4382. (13) Turner, D. H.; Flynn, G. W.; Sutin, N.; Beitz, J. V. J. Am. Chem. Soc. 1972,94,1554. (14) Hams, D.C. Quantitative Chemical Analysis; W. H. Freeman and Company: San Francisco, 1982;p 384. (1 5) Swift, E. H.; Butler, E. A. Quantitative Measurements and Chemical Equilibria; W. H. Freeman and Company: San Francisco, 1972;p 374. (16)Vogel, A. I. Textbook of Quantitative Inorganic Analysis, 3rd ed., Wiley, New York. 1961,p 265.
6331
(17)Gcffith, R. 0.; Mckwwn, A. Trans. Faraday Soc. 1935, 31, 868. (18) Eigen, M.; Kustin, K. J . Am. Chem. Soc. 1%2,84, 1355. (19) Wang, Y.L.;Nagy, J. C.; Margerum, D. W. J . Am. Chem. Soc. 1989,111,7838. (20)Simoyi, R. H.; Eptein, I. R. J. Phys. Chem. 1987,91,5124. (21)One of us (KK) observed that the hydrolysisof thiocyanogen was t w fast to measure even by temperature-jump techniques. (22) Kim, J. L.;Cleveland, J. P. J . Am. Soc. 1973.95, 104. (23)Gerritsen, C.M.; Margerum, D. W. Inorg. Chem. 1990,29, 2757. (24) Kap, P.;Rentrop, P. Numer. Math. 1979,23, 5 5 .
Kinetlcs and Identlflabillty of Intramolecular Two-State Exclted-State Processes. Global Compartmental Analysis of the Fluorescence Decay Surface Nod Boens,**tR o n Andriessen,? Marcel Amelooft Luc Van Dommelen,? and Frans C. De Scbryvert Department of Chemistry, Katholieke Universiteit Leuven, B-3001 Heverlee- Leuven, Belgium, and Limburgs Universitair Centrum, B-3590 Diepenbeek, Belgium (Received: November 21, 1991; In Final Form: March 12, 1992)
This report discusses the fluorescence decay analysis of intramolecular two-state excited-state processes in terms of compartments. The kinetic equations specifying the fluorescence decay and the time course of the two _excited-statespecies concentrations are derived in terms of the rate constants kti, and the spectroscopic parameters b, and E l . bl and El are respectively the relative absorbance and the normaked spectral emusion weighting factor of species 1. The structural identifiability study demonstrates that at least three parameters of the intramolecular two-state excited-state process must be known in order to recover the rekvant information. These three parameters can be (a) two rate constants and one spectroscopic parameter [( 1) 2 k,, and 1 b l , (2) 2 k,, and 1 E l ] ; (b) one rate constant and two spectroscopic-parameters [(3) !k,, 1 b l , and 1 5 , (4) 1 k, and 2 gl, ( 5 ) 1 k, and 2 E l ] ; and (c) three spectroscopic palameters [(6) 2 bl and 1 E l , (7) 1 bl and 2 E l , (8) 3 bl, (9) 3 EA. The conditions (3), (6), and (7)require that the known (b,,Z.,)values do not belong to a decay trace where the preexponentials al,azhave the same absolute values but opposite signs ( q / q= -1) corresponding to (bl,fl) = (1,O) or (0,l). Conditions (6) to (9) indicate that spectroscopic information alone can suffice for the system to be identifiable. Finally, the unknown parameters are estimated from computer-generatedfluorescence decay traces. The results confirm the conclusions of the identifiability study.
1. Introduction The kinetics of excited-state reactions can be characterized from time-resolved measurements of the concomitant fluorescence emission. Current methodologies in time1**and frequency domain3*4provide data with sufficient accuracy allowing an elaborate data analysis. In many cases the time relaxation of the excited system can be described by a sum of exponentially decaying functions. The parameters of interest are not the relaxation times and the correspondmg preexponential factors (these may be called descriptive or empirical parameters) but the rate constants in the excited state and the species associated spectra. To determine these parameters of interest experiments have to be carried out under a variety of conditions so that a multidimensional fluorescence decay data surface is obtained. From the resulting set of relaxation times and preexponential terms the rate constants and the relative species associated spectra can be obtained. An accurate determination of the relaxation times and the preexponentials and a high model discrimination power is obtained by performing a simultaneous analysis of all related decay trace^.^-'^ In this so-called global analysis approach parameters can be linked over various decay curves. For example, the relaxation times can be linked Over decay traces collected at various emission wavelengths. However, for decay curves collected at various concentrations of a reactant the relaxation times can generally not be linked. To benefit the most *Towhom correspondence should be addressed.
t Katholieke Universiteit Leuven.
* Limburgs Universitair Centrum. 0022-3654/92/2096-633 1$03.00/0
of the merits of a global analysis approach, one has to fit for the underlying parameters, i.e., the rate constants which can be linked indeed. This global analysis has the additional advantage that the parameters of interest are determined directly from the complete decay data surface in a single step.+I5 The expressions for the relaxation times and the preexponential factors in terms of the rate constants and the species associated spectra for a causal, linear, time-invariant, and concentration-dependent fluorescent system consisting of n different excited-state species can be obtained in matrix notati~n.l'-'~In the first implementation of this direct approach the decay curves had to be properly normalized to obtain the species-associatedspectra." Recently, an extension of this method has been described in which the condition for normalization is no longer required.I3 Besides the problem of the determination of the parameters of interest, the following question can be raised: what are the type and the number of experiments necessary to identify these parameters. This identifiability problem has been discussed for the case of concentration-dependent excited-state reactions1*J3using the approach described for compartmental analysis.I6 The mathematical theory of the kinetic behavior of systems of compartments is called compartmental analysis, and it has been found useful for the analysis of experiments in many branches of the bio~ciences.'~J' In this paper the compartmental description and the identifiability of intramolecular two-state excited-state processes will be discussed. A separate study of intramolecular excited-state reactions is justified by the fact that for these processes the concentration axis cannot be used in contrast to the previously discussed intermolecular excited-state reactions. The results of this 0 1992 American Chemical Society
6332 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 SCHEME I
Boens et al. If the system depicted in Scheme I is excited by a &pulse, the concentration dependence of the two excited species 1* and 2* as a function of time is described by the following system of differential equations: d[l*]/dt = -(kol
t I k021
I
(la)
hv
t I
I t
U
U
study can also be used for the intermolecularreactions when the explicit concentration dependence is not known. First, the kinetics of the fluorescence decay of intramolecular two-state excited-state processes upon delta pulse excitation will be described. Second, the structural identifiability problem to be solved is whether or not the system parameters are uniquely defined under error-free conditions. This problem reduces to the question of whether or not an algebraic system of nonlinear equations p0wxe.s a unique solution. This investigation will yield precise requirements for the system to be identifiable. Third, the results of the identifiiability study will be verified by the estimation of the unknown parameters from computer-generated fluorescence decay traces.
2. Compartmental Systems In photophysics a compartment is composed of a distinct type of species which acts kinetically and spectroscopically in a unique way. The concentration of the constituting species can change when the compartments exchange material through an intramolecular or intermolecular process. Compartments can be divided into ground- and excited-state compartments depending upon the state of the composing species. An excited-state system can be viewed as a compartmental system made up of a number of excited-state compartments. There may be inputs from the ground-state compartments into one or more of the excited-state compartments by light excitation. There is output from the excited-state compartments to the ground-state compartments by deactivation such as fluorescence, internal conversion, .... If the concentrations of the species in the ground state are not substantially altered upon excitation, it suffices to consider only the excited-state compartments. In that case, the photophysical system can be regarded as open. A compartmentalsystem can be closed by adding the ground-state compartments corresponding to the excited-state compartments. Compartments are depicted as boxes (see Scheme I), circles, or ovals which enclose the composing species. Single-headed arrows pointing away from a compartment are used to represent outflow from that compartment, whereas single-headed arrows pointing toward a compartment depict inflow into that compartment. 3. Kinetics
Consider a causal, linear, time-invariant, concentration-independent (i.e., intramolecular) system consisting of two different types of excited-state species (i.e., a bicompartmental excited-state system). The kinetic model for such a system is depicted in Scheme I. Species 1 can undergo a reversible transformation to species 2. Excitation by light creates the two different excited-state species 1* and 2*, which can decay by internal conversion (IC), fluorescence (F),and intersystem crossing (ISC). The rate constants for those processes are denoted by kol (=kFi + klcl + kBcl) and kO2(=kn + klc2+ kIsc2). The rate constant describing the transformation 1* 2* is represented by kzl and for the transfer 2* 1* by kI2. Note that all rate constants k, are nonnegative.
-
+ kZl)[l*] + k12[2*]
-
(3)
(4) r
1
(5)
df
we can write the differential equations (1) in matrix notation as
X*'(t) = AX*(t), t 2 0 and the initial conditions (2) as X*(O) = b
(6)
(7) The 2 X 2 constant matrix A [aij]is called the compartmental matrix. This matrix has the properties that (1) each off-diagonal element is nonnegative, (2) each diagonal element is nonpositive, (3) the sum of any column, say thejth column, equals -koP In the terminology of systems theory the matrix A is also referred to as the system matrix. The elements b, of the 2 X 1 column vector b are related to the ground-state absorbances of species i by
where el and t2are the molar extinction coefficients of 1 and 2, respectively. The solution of the system of differential equations (1) or (6) is given byi8
X * ( t ) = exp(tA)b
(9)
with exp(tA) defined as exp(tA) = I
+ E-(tA)n n! "-1
where I is the identity matrix. Usually it is preferable to compute exp(tA) by using eigenvalues rather than attempt to sum the series (10). The Compartmental Matrix Has Two Distinct Eigenvalues. Assume that the 2 X 2 compartmental matrix A has two linearly independent eigenvectors PI and P2 associated with the eigenvalues y I and y2,respectively, Le., A = P r P 1with P = [PirP2]and Pi the inverse of the matrix of the eigenvectors, and that the matrix r = diag(yl,y2). Equation 9 can then be written as
X*(r) = P exp(tI')Pib (11) where exp(rr) = diag(exp(y,t), exp(y2t)). P and exp(tr) are functions of the rate constants hl,k21,kO2,and kI2. b is dependent on the excitation wavelength A"". Equation 11 can explicitly be expressed as a sum of exponentials [1*1(0 = 811 exp(y1t) + 812 exp(y2t) (1 2a) [2*l(t) = 8 2 1 exp(y1t) + 8 2 2 exp(y2t)
( 12b)
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6333
Intramolecular Two-State Excited-State Processes The exponential factors yi are related to the decay times ri according to Yi
= -1/71
\
3.8
(13)
and are given by 71.2
+ X2
= -'/{Xi
[(XI - X2)'
+ 4kzik121~'~) (14)
with XI = k01 + k21
(153)
xz = ko2 + k12
(15b)
811
812 821
0
- b2k12I/(YZ - 71)
( 16a)
= -MX1 + 71) - b2k121/(72 - Y l )
(16b)
= lb2W2 + Y2) - blk21)/(Y2
(16c)
= @,(XI + YZ)
-Y
1.81 1 6 1
The preexponential terms are
- Yl)
200
400 time (arbitrary units)
200
400
= ib2(X2 + 71) - blk211/(Y2 - 71) (16d) If k12= 0, the time dependence of the concentrations of excited species 1* and 2* is given by 822
[1*l(t) = bl exP(Ylt)
(17a)
0.5 !
0
I
time (arbitrary units)
YZ = -ko2
(18b)
blkZl/(Yl - 72)
( 19a)
and 821 = 822
= 62 - blkZl/(71 - 72)
( 19b)
The decay of [1*] is monoexponential while that of [2*] is biexponential. However, depending on the value of 821and PZ2 we can categorize the biexponential decays (q17b) into three separate classes: (1) If y1> y2 ( r 2C rl), Le., ko2> kol + kll or X2 > X I , and b2 C b l k 2 1 / ( ~-1 r2), Le., bZ2C 0, then the decay of [2*] is biexponential with a rise (negative preexponential term)and decay (positive preexponential term) component. The decay component has the same decay time (T~-') as the monoexponential decay of [1*]. This is illustrated in Figure la. (2) If yI > 7 2 (72 < r l ) ,bz > b1kZ1/(YI- y2), i.e., Oz2 > 0, then the decay of [2*] is a sum of two-exponential terms. One of the decay times (-yI-l) is the same as that of the monoexponential decay of [ 1*I. This is shown in Figure 1b. (3) If y I C y2 ( r l C T2)r i.e., kO2C kol + kzl or X2 C XI, the decay of [2*] is biexponential with a rise C 0) and decay component. The rise component with the negative preexponential term has the same decay time as the monoexponential decay of [ 1*I. This is illustrated in Figure IC. For cases 1 and 3 the negative preexponential term is always associated with the shorter decay time.lg In the complementary case, when k21= 0 the time dependence of the concentrations of excited species 1* and 2* is given by eqs 17-19 with interchanged indexes 1 and 2. The explicit expressions are given below for the convenience of the reader. [I*l(t) = 811 exP(Ylt) + 8 1 2 exP(Yzt) Goa) P*I(t) = 82 exP(Yzt)
" , 0
460
200
time (arbitrary units)
Figure 1. Time dependence of the concentrations of the excited species 1* and 2* when either kI2= 0 or kll = 0 (Scheme I). See text for more details.
(22b) = b2k12/(Y2 - 71) If k12= 0 and kZl= 0, the concentrations of excited species 1* and 2* decrease monoexponentially with time and are given by [l*l(t) = bl exP(Y1r) (234 812
P*l(O = b2 exp(72t)
(23b)
with Gob)
Y1
= -ko1
(24a)
YZ
= -k02
(24b)
with Yl 72
= -ko1
= -(koz
+ k12)
The Compartmental Matrix Has One Eigenvalue with Multiplicity Two. If the 2 X 2 matrix A has one distinct eigenvalue
6334 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 y
with multiplicity two, X*(t) is given by'* X * ( t ) = exp(yt)[I + t(A - yI)]b
(25) The explicit expressions of eq 25 are given in the supplementary material (see Supplementary Material Available paragraph at the end of this article). Fluorescence Decay Kinetics. In fluorescence decay experiments, one does not observe X * ( t ) directly, but the composite spectral emission contours of the excited-state species. The fluorescence &response function, flXCm,XCx,t), at emission wavelength XCm due to excitation at Xcx is expressed by flXcm,Xcx,t)
= cX*(t) = c(Xcm)exp(tA)b(Xex), t 1 0
kFiis the fluorescence rate constant of species i; pi(Xcm) is the spectral emission density of species i at emission wavelength Xem, normalized to the complete emission band, and AXCm is the emission wavelength interval in which the fluorescence is monitored. pi(Xcm) is defined by = F,( Acm) /
1
full band
wavelength whereas &(Acx) depends on the excitation wavelength only. In global compartmental analysis one fits directly for the rate constants kij, the normalized spectral emission wighting factors E and the normalized zero-time concentrations b of the two ground-state species. The steady-state emission spectrum, F,, in terms of A, b, and E is obtained by integration of eq 34
= -KEA-% where A-l is the inverse matrix of A,
F,( Xcm) dXem
If the compartmental matrix A has two distinct eigenvalues y, and y2 with corresponding eigenvectors PI and P2, respectively, eq 34 can be formulated as flXcm,Xex,t) = KE(P)P exp(tr)Pl&(Xcx), t 2
fi(Xcm,Xex,t) = ~l(X~'")[l*](t)
= CZ(XC") [2*](t)
= cz(XCm)[ & 1 ( 1exp(y10 ~~) + B22(Xcx) exp(rzt)l (29b)
4. Species-Associated Emission Spectra (SAEMS)and Species-Associated Excitation Spectra (SAEXS) Xcm, due to excitation at Xcx, F,(XCm,XCx), can be written as13 2
Fs(Acm,Xcx) = p EEi(Xcm)[i*]s(Acx) i= I
(30)
exp(71t) + [clB12+ c2B221exp(y2t) which can formally be expressed as =
(38)
where p is a proportionality factor. [i*], is the steady-state value of [i*](t) given by (39)
The contribution of species i to the steady-state emission spectrum F,(Xm,XQ) will be called the species-associated emission spectrum of species i, SAEMSi(XCm,XCX), SAEMSi(Xcm,XCX) = pEi(Xcm)[i*],(Xcx)
with Bij defined by eq 16. Equation 26 can be written as
+ f2(Xcm,Xex,t)
(37)
P exp(tI')P1 is a function of the rate constants k, only.
[i*], = x m [ i * ] ( t )dt = -(A-lb)i
flf(XCm,XCX,t)= fi(Xcm,Xcx,t)
o
The steady-state fluorescence spectrum at emission wavelength
where F, denotes the steady-state fluorescence spectrum. If the 2 X 2 matrix A has two distinct eigenvalues y l and y2, the time-dependent fluorescence of each species,f;.(t), detected over the emission band AXCmcan be expressed by
f2(Xc",hcx,t)
(35)
~,(~em,~cx)
(26)
where c is the 1 X 2 vector [cl c2]of spectral emission weighting factors ci( A'") expressed by
p i ( Xcm)
Boens et al.
(40)
or equivalently, by using eq 35
~ B I+I~ $ 2 1 1
flXcm,XcX,t) = a1exp(ylt) + a2 exp(y,t),
t 20
(31)
with aithe ith preexponential. If the compartmental matrix A has two distinct eigenvalues yl and yz,the fluorescence &response function of the system depicted in Scheme I is biexponential with decay times si = -l/yi independent of hexand XCm. From eqs 29-31 it is clear that the preexponentials ai depend on the rate constants kijand on the excitation and emission wavelengths. The information about the equilibrium in the ground state specified by bl ([l*](O)) and b2 ([2*](0)) can be assessed by an elaborate mathematical procedure from the ratio of the preexponentials. The same is true about the information about the spectral emission weighting factors cl and c2. If the elements bi of b are normalized as hi
= bj/Dj
(32)
and similarly the elements ci of c as Ej
=c p j
We have shown in the section on kinetics that when kzl = 0 and k12= 0 the decays of 11'1 and [2*] are both monoexponential with decay constants y i = -l/si defined by eq 24. In that case, eqs 31 and 34 at time t = 0 yield Crl CY2 = K ( E l 6 1 + ? 2 6 2 ) (42)
+
Equation 35 and integration of eq 31 between the limits t = 0 and t = lead to LYlk02
+ a2kol = K(fl61k02 + E262k01)
(43)
From eqs 42 and 43 we have that &I
= KElhl
(444
(Y2 = KE$2 (44b) and the species-associated emission spectrum of species i, SAEMSi(XCm,XCX), is then
CtjTj
(33)
SAEMSi(XCm,XCX) = -F,( Xem,Xex) cajsj
(45)
J
eq 26 can be written as flXCm,XCX,t) = K E ( X ~ exp(rA)&XCX), ~) t 1 0
(34)
with K a proportionality constant. The use of K , 6,, and Ei allows one to link Ei over decay curves collected at the same emission wavelength and to link bi over decay curves obtained at the same excitation wavelength. Indeed, E(Xm) depends just on the emission
Equation 45 is the formula derived by Wahl and Auchetmfor the so-called decay-associated spectra (DAS). Thus the decay-associated spectra become the species-associated emission spectra when there is no excited-state process between the excited species 1* and 2*. The species-associated excitation spectra, SAEXS, can be o b tained in a similar way as for the SAEMS. If Es(Xm,Xcx) denotes
Intramolecular Two-State Excited-State Processes
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6335
the steady-state excitation spectrum of the sample recorded at XCm,SAEXSi(XCm,XCX) for species i is given by
with 6, = 1 - 6, and E2 = 1 - E l . Iff(t) is a sum of exponentials (eq 31), the Markov parameters can also be represented as 2
mi = Ca,?;, i = 0, 1
(57)
i
5. Identifiability of Intramolecular Bicompartmental Systems In this sectio? we shall discuss whether or not the system parameters kij, b,, and El can be uniquely determined from the experimental measurement of the fluorescence decay surface of intramoleculartwestate excited-stateprocesses. The kinetic model is described by the differential eqs 6 and the initial conditions defined by eq 7. The fluorescence &response functionf(Xcm,Xex,t) is given by eq 26 or 34. For the system depicted in Scheme I the fluorescence decay trace d,( XCm,XCX,t) measured at emission wavelength XCmdue to excitation at Xcx is the convolution off(t) and the instrument response function u(Xcm,Xex,t):
where 7; denotes the ith power of rj. For a decay trace collected at emission wavelength Xim due to excitation at A?, elimination of the scaling factor K from eqs 55 and 56 leads to E1[6,(P - Q)
+ Q]
6l(R - S )
+S
(58)
with
P
ml
+ mo(kol+ 2k2J
Q -ml - mo(ko2+ 2kI2)
(59) (60)
S = -ml- mo(ko2+ k I 2 )
= dEx' exp[(r - r?A]bu(t? dt'
= d[E exp(tA)b] @ u(t)
(47)
in which @ denotes the convolution operator, and the scaling factor d is proportional to K . From eq 47 it is clear that all system parameters must be identified through the &response functions At). The problem of structural identification of the model can now be described as follows: is it possible to uniquely determine the unknown parameters ky, and El from knowledge of the fluorescence &responsefunctionsf(t)? The closely related problem of actually estimating numerical values of these unknown parameters from the measured fluorescence decay surface (i.e., the parameter estimation problem) will be considered in section 6. Let us examine what formulas are available relating the parameters of the fluorescence 6-Eesponse function f ( t ) to the unknown system parameters kii' b,, and E l . First, information is available from the characteristic polynomial of A which can be expressed as det (71 - A)
h(r - T i )
I= I
y 2 - aly + u2
CYi i
02
= CYiYj i
