J. Phys. Chem. 1995,99, 8959-8971
8959
ARTICLES Distinction between Different Competing Kinetic Models of Irreversible Intramolecular Two-State Excited-State Processes with Added Quencher. Global Compartmental Analysis of the Fluorescence Decay Surface Luc Van Dommelen, Nobl Boens,* and Frans C. De Schryver Department of Chemistry, Katholieke Universiteit Leuven, B-3001 Heverlee, Belgium
Marcel Ameloot Limburgs Universitair Centrum, B-3590 Diepenbeek, Belgium Received: October 26, 1994; In Final Form: March IO, 1995@
In this report we investigate how, even in the absence of any a priori information, one can distinguish between competing irreversible intramolecular two-state excited-state processes. Only two different kinetic models are possible for irreversible intramolecular two-state excited-state processes: (i) one with a unidirectional excited-state process and (ii) one without an excited-state process. The distinction between these models is based on the collection of a fluorescence decay surface with at least three quencher concentrations and the use of standard global biexponential and global compartmental analyses. Standard global biexponential analysis provides estimates for the preexponential factors and decay times which, together with the steady-state fluorescence spectrum, allow the construction of decay-associated emission spectra. Two series of global compartmental analyses have to be performed whereby (i) k01 is kept constant at various preset values while k12is held fixed at zero and whereby (ii) k 0 2 is scanned while k21 is held constant at zero. kol denotes the rate constant of deactivation of excited species i*, klz represents the rate constant of transformation of excited species 2* into 1*, and k21 describes the opposite process. If negative preexponentials-indicative of a unidirectional excited-state process-are obtained in the global biexponential analysis, the statistical goodnessof-fit criteria of the repetitive global compartmental analyses allows one to determine the direction of the unidirectional excited-state process. When only positive preexponentials are obtained in the global biexponential analysis, the decay-associated emission spectra should be compared to the species-associated emission spectra. If both sets of spectra coincide, no excited-state process occurs. Otherwise, two solutions in which one rate constant of interconversion is zero are mathematically possible. It is impossible to distinguish between these two nontrivial alternative solutions.
Introduction Time-resolved fluorescence spectroscopy is a powerful technique in unraveling the often complex kinetics of excitedstate processes.'.* In many cases, the time dependence of the fluorescence can be described by a set of time-independent rate constants and spectral absorption and emission parameteix3-I6 To estimate numerical values of these so-called system parameters, fluorescence decay traces are collected under various experimental conditions, resulting in a multidimensional fluorescence decay surface. The global compartmental analysis of all related experimental fluorescence decay traces is very efficient in distinguishing between competing kinetic models and in obtaining reliable estimates of the system parameters in a single ~ t e p . ~ - ' ~ To fully characterize reversible intramolecular two-state excited-state processes in the absence of added quencher, three system parameters have to be known beforehand.l0 When no a priori information on the rate constants of intramolecular twostate excited-state processes is available, limits on these rate constants can still be specified if the fluorescence decay surface
* To whom correspondence should be addressed. @Abstractpublished in Advance ACS Abstracts, May 1, 1995.
includes a decay trace whereby only one species is excited and the fluorescence of the corresponding excited-state species is monitored.I5 When at least three different quencher concentrations (one of which may be equal to zero) are used, only one system parameter, which is not a rate constant of quenching, has to be known a priori.13 In this case, the two rate constants of quenching must be different. Limits on the rate constants of reversible intramolecular two-state excited-state processes can be specified even when no a priori information is accessible.I6 This method also allows one to distinguish reversible from irreversible intramolecular two-state excited-state processes in the absence of a priori information. In this paper we will investigate how to distinguish between the different possible kinetic models for irreversible intramolecular bicompartmental systems with added quencher in the absence of a priori information, i.e., (i) how to distinguish between competitive intramolecular bicompartmental systems with a unidirectional exchange between the excited-state compartments, and (ii) how to make a distinction between intramolecular bicompartmental systems with a unidirectional exchange from those with no exchange between the excitedstate compartments. For a definition of the term compartment in photophysics, see ref 15. The conclusions of the theoretical
0022-3654/95/2099-8959$09.00/0 0 1995 American Chemical Society
8960 J. Phys. Chem., Vol. 99, No. 22, 1995
Van Dommelen et al.
SCHEME 1
SCHEME 2
-
A
-
k21 I
I
section are investigated by global compartmental analyses of computer-generated fluorescence decay surfaces.
'*
B
Theory
A. Fluorescence Decay Kinetics. Consider a causal, linear, time-invariant, concentration-independent (Le., intramolecular) system consisting of two distinct types of excited-state species with added quencher, as depicted in Scheme 1. Ground-state species 1 may reversibly transform into ground-state species 2. Excitation by light creates the excited-state species 1* and 2* which can decay by fluorescence (F), internal conversion (IC), and intersystem crossing (ISC). The composite rate constants describing these processes are denoted by IQI (= ~ F I krc~ k1sc-J. The rate constant k ~ s c l )and ko2 (= k ~ 2 k1c2 describing the transformation 1* 2* is represented by k21, whereas the transformation 2* 1* is characterized by k12. In this paper we will deal with irreversible systems (Le., systems with k21 and/or k12 equal to zero). Three kinetic models as depicted in Schemes 2 and 3 are possible for such irreversible systems. The added quencher Q accelerates the depopulation of the excited states of species 1* and 2* by kql[Q][l*] and k~[Q][2*], respectively. For all considered cases, it is assumed that Q affects exclusively the excited-state species deactivation and does not change in any way the ground-state equilibrium. Furthermore, the quenching is assumed to be of the SternVolmer type (Le., only time-independent quenching rate constants are considered). If the systems depicted in Schemes 1-3 are excited by a d-pulse which does not significantly alter the concentrations of the ground-state species, the fluorescence 8-response function, f(lbem,Aex,t), measured at emission wavelength Aem due to excitation at Lexis expressed by6
+
+
+
+
--
with K being a proportionality constant. U = [VI, U2] is the matrix of the two eigenvectors of matrix A, and U-' the inverse of U. yl and y2 are the eigenvalues of A corresponding to U1 and U2, and expttr) = diag{exp(ylt), exp(y2t)). The compartmental matrices A according to Schemes 1, 2 , and 3 are given by eqs 2, 3, and 4,respectively.
SCHEME 3
b is the 2 x 1 vector of normalized absorbances 6; of species i at Aex.lo b depends on lex.
J. Phys. Chem., Vol. 99, No. 22, 1995 8961
Distinction between Competing Kinetic Models f is the 1 x 2 vector of the normalized emission weighting factors Z.i of species i* at Aem.l0 e depends on A"".
flilem,ilex,t)= K[B,z, exp(y,t)
+ (1 - 6,)(1 - E,) exp(y201,
t 2 0 (13)
with Equation 1 can be written in the biexponential format:
j?Jem,ilex,t)= a,exp(y,t)
+ a,exp(y2t),
t20
Yi = (7)
The preexponential factors ai are dependent on kot, k21,ko2, k12, kQt, kQ2, [QI, 61,and Z.I. The exponential factors yi are relatedto the decay times 'ti according to
yi = -1Iq
(8)
and are a function of kQl, k ~ 2[Q], , and SI,S2, and P are defined asI6 Sl
= k01 + k21
(94
-~Q,[QI
(14)
and y2 is given by eq 12b. In that case, only a sum of two exponential terms is possible for jfAem,Aex,t). B. Specifying Rate Constant Bounds. B.l. Kinetic Model with k21 f 0 and kl2 # 0 (Scheme 1). Although a reversible ( P f 0) intramolecular bicompartmental system with added quencher is not identifiable without additional information on the system parameters, one can specify limits on the rate constants using the numerical values of SI, S2, and P obtained through the so-called scanning procedure.I6 Scanning one of the rate constants (hl,k21, h2,k12) means that this rate constant is kept constant at various preset values during successive global compartmental analyses. The bounds on the rate constants are given by
0 < kol < S, - PIS2
(154
PIS, < k21 < S ,
( 15b)
0 < kO2< S2 - PIS,
(15c)
PIS, < k , , < S2
(1 5 4
p = k21ki2 For the kinetic models according to Schemes 2 and 3, the parameters SI and S2 can be simplified. The parameter P = 0 for all kinetic models, according to Schemes 2 and 3. If k12 = 0 (Scheme 2A), AAemP,t)is given bylo
8.2. Kinetic Models with k12 = 0 or k21 = 0 (Scheme 2). In this case, the intramolecularbicompartmental system is irreversible ( P = 0). The use of the scanning p r ~ c e d u r e ' ~whereby -'~ the rate constant kol (or k21, Scheme 2A) is kept fixed at different preset values during successive analyses allows one to set limits on kol and k21 (eq 16). The values of kol and k21 are related through eq 9a.
with
-= S ,
( 16a)
0 < k21 < SI
(16b)
0 < k,,
Two cases can be distinguished, depending on the relative magnitude of YIand y2: (i) Y I > y2 (TI > n),i.e., ko2 + kAQ1 > k o ~ k21 k Q l [ Q ] . If 61 > ( y ~- y2Myt - y2 k d , the is biexponential fluorescence &response function AAem,Aex,t) with a rise (negative preexponential factor associated with the short decay time ' t 2 ) and decay (positive preexponential factor) component. If 61 < ( y l - y2)/(y1- y2 k21),AAemP,t)is a sum of two exponential terms. As y l and y2 depend on [Q], the switch of AAemP,t)from a sum to a difference of two exponential terms will occur at different 61 values for different [Q]. If 61 = 1, AAemP,t)is always a difference of two exponential terms. (ii) yl < y2 (TI < zz), Le., k o 2 kQdQ1 < k01 k21 kq1[Q1. If Z.1 < k21/(y2- y~ k d , A A e m P , t )is a difference of two exponential terms. A sum of two exponential terms is found for AAem,Aex,t) if .?I > k214y2 - 71 k21). Note that, as yl and y2 depend on [Q], the switch of AAemP,t)from a sum to a difference of two exponential terms will depend on [Q]. Since y l and y2 depend on [Q], it is possible to switch from case i to case ii as a function of [ Q ] . If k21 = 0 (Scheme 2B),flAemJex,t)is given by eqs 11 and 12 with interchanged indexes 1 and 2. Whether a sum or difference of two exponential terms will be obtained for fiAemJex,t) can be deduced from the previous discussion by interchanging the indexes 1 and 2. If k21 = 0 and k12 = 0 (Scheme 3),AAemP,t) is given by
+
+
+
+
+
+
+
+
+
Since P = 0, it follows that k12 = 0 and so k o 2 = S2. For irreversible intramolecular two-state excited-state processes specified by a particular set of {SI,S2, kql, kQ2) values, an altemative solution is theoretically possible if the rate constant k02 (or kl2) is scanned instead (Scheme 2B). In this case, the rate constants ko2 and kl2 are limited between zero and S2 (eq 17) while kol (= St) and k21 (= 0) are determined exactly. The values of k o 2 and k12 are related according to eq 9b.
0 < kO2< S2
( 17a)
0 < k , , < S2
(17b)
It should be emphasized that the numerical values of SI, S2, kQ1, and kQ2 are identical for Schemes 2A and 2B. The same rate constant values of quenching, k ~ and l kQ2, are obtained as in the case where k~l(or k21) was scanned so that the numerical values of the combinations (SI,kQl) and (S2, kQ2) are still preserved. This means that scanning of the rate constants k01 (or k21, Scheme 2A) and k02 (or k12, Scheme 2B) for the same ) theoretically two numerical values of {SI,s2, kQ1, k ~ 2 gives different mathematical solutions for the same data surface. This altemative solution is not the trivial one obtained by switching the labels of the compartments.16 In the section Results and Discussion we will further investigate the conditions under which this altemative solution can be found.
8962 J. Phys. Chem., Vol. 99, No. 22, 1995
Van Dommelen et al.
B.3. Kinetic Model with k21 = k12 = 0 (Scheme 3). According to the kinetic model of Scheme 3 (k21= k12 = 01, the following values for the rate constants are found: k~land ko2 are equal to SI and S2, respectively. In this case, all the rate constants can be exactly determined by the scanning procedure. C. Species-Associated Emission Spectra (SAEMS). The contribution of species i* to the total steady-state fluorescence spectrum F(Aem,Aex)at emission wavelength A'" due to excitation at ;lexis called its species-associated emission spectrum, SAEMSi(Aem,Aex), and is given by6
SAEMS,(Z",A"") = Qi(Tm,Aex) F(Zm,Zx) (18a) with
Q l = [Zl(Aem)(A-'b(AeX))I]/[E(Aem)A-lb(AeX)] (18b)
xi, subject to constraints on the
minimizing the global reduced values of the fitting parameters: U
where the index 1 sums over q experiments and the index i sums over the appropriate channel limits for each individual experiment. y: and yyl denote respectively the observed (synthetic) and calculated (fitted) values corresponding to the ith channel of the lth experiment, and wll is the corresponding statistical weight. Y represents the number of degrees of freedom for the entire multidimensional fluorescence decay surface. It is crucial that all fitting parameters are subject to simple range constraints on their values. The problem of minimizing 2: can be stated mathematically as follows:
with A-I as the inverse of the matrix A. If and only if k21 = = 0 (Scheme 3), thenlo
minimize &)
k12
for all x,
subject to sj 5 xJ 5 tJ D A S ~ ( A ~ ~=, A [a,(iem,~ex).rjCa,(Aem,~ex)~~F(Aem,Aex) ~~) = SAEMS,(A'",A'")
(19)
where DASi(Aem,Aex) is the decay-associated emission spectrum of species i*." D. Distinction between Different Kinetic Models of Intramolecular Two-State Excited-State Processes. Scanning a particular rate constant makes it possible to distinguish reversible from irreversible intramolecular two-state excitedstate processes.I6 If P (eq 10) # 0, both rate constants k21 and k12 have nonzero values and the system is said to be reversible (as depicted in Scheme 1). This system has been investigated in detail elsewhere.I6 If P = 0, the system is called irreversible. Two competing kinetic models are possible in this case (Schemes 2 and 3). Negative preexponential factors for a system with P = 0 are indicative of a unidirectional excitedstate process (Scheme 2 is valid). If biexponential decay traces with only positive preexponentials are included in the fluorescence decay surface and if P = 0, it is impossible to know beforehand whether there is a connection between the excitedstate compartments. In this case, a distinction between the two competing kinetic models (Schemes 2 and 3) can be made by comparing the DAS to the SAEMS. If the DAS are different from the SAEMS, the correct kinetic model is represented by Scheme 2. Otherwise, Scheme 3 represents the correct kinetic model.In For the systems depicted in Scheme 2 , two different solutions (apart from the trivial switching of the labels of the compartments) are mathematically possible (Schemes 2A and 2B). Scheme 4 presents a flowchart depicting the process of distinguishing between the different kinetic models of intramolecular two-state excited-state processes. The possible routes in this process will be fully discussed in the Results and Discussion section. Parameter Estimation A. Program Implementation. The global compartmental analysis of the fluorescence decay surface of species undergoing excited-state processes was implemented in the existing general '~ global analysis programls based on M a r q ~ a r d t ' s algorithm. For intramolecular two-state excited-state processes with added quencher, the global fitting parameters are kol, k21,kQl, kO2, klz, k ~ 2 hl(kex), , and ?](Aem). A detailed description of the program implementation of global compartmental analysis has been given elsewhere.I3 The fitting parameter values were estimated by
x E R"
(21)
j = 1, 2 , ..., n
with n the number of adjustable parameters. This format assumes that upper and lower constraints exist on all fitting parameters. Restrictions on the values of a particular fitting parameter j can be removed by allowing very large negative and positive values for respectively s, and t,. For all rate constants, s, was set at -0.01 ns-I; for the local scaling factors (see Scheme 5) s, was set at 0. The default constraints on 81 and ?I are -0.5 5 bl, Z.1 5 1.5. Small negative s, prevent oscillations in the nonlinear least-squares search which would occur if the values of the fitting parameters were forced to be non-negative. These constraints have to be set differently in some analyses (see Results and Discussion). The numerical statistical tests incorporated the calculation and its corresponding Z7?, of
xi
Using Zx2,the goodness of fit of analyses with different v can be readily compared. For simulations, fits with Zx2> 5 E are considered to be unacceptable. The additional statistical criteria to judge the quality of the fits are described elsewhere.2 To determine the intervals of the rate constants k,, series of global compartmental analyses were performed in which (at least) one of the rate constants k, was kept constant at different preset values, while the remaining fitting parameters were freely adjustable (i.e., the scanning procedure). The decay parameters were linked as shown in Scheme 5. Boxed parameters are linked, while X denotes the local scaling factors. B. Synthetic Data Generation. Synthetic sample decays were generated by convolution of A;lem,Aex,t) with a nonsmoothed measured instrument response function. The preexponential factors al,2and corresponding eigenvalues y1.2 of the biexponential decays were computed from the rate constants k,,,kQl, kQ2, 81, E , , and [Q] by a dedicated computer program. The preexponential factors were adjusted to obtain the desired number of counts. All computer-simulated decays had '/zK data points with lo4counts in the peak channel. The time increment per channel was chosen to ensure that the final decay intensity was about 5% or less of the peak intensity. Full details of the decay data simulations are given elsewhere.zn In the tables, a value followed by (c) means that this parameter was kept constant at the given value during the
J. Phys. Chem., Vol. 99, No. 22, 1995 8963
Distinction between Competing Kinetic Models
SCHEME 4
See Van Dommelen et al., J. Phys. Chem., 97,11738 (1993)
N~ 1. scanning kO1 with k12= 0 constant ...................................................................... 2. scanning kO2with k21 = 0 constant I
1
1. scanning kO1 with kI2= 0 constant
I 2. scanning kO2 with kpl = 0 constant I r C o e e DASISAEMS
I
Scheme Ila
Scheme 111 SAEMS
yes
e 3 Scheme II
analysis. The true (i.e., simulation) values are indicated by the superscript t. All quoted errors are 1 standard deviation.
Results and Discussion
A. Systems with k12 = 0 and k21 t 0. In this section, three different fluorescence decay surfaces are considered. The first decay surface consists exclusively of decay traces whereby flAemp,t) is a difference of two exponential terms. In the second decay surface, only decay traces with positive preexponential factors are included. Finally, the third decay surface has decays wherebyflAemp,t)can be a sum or difference of two exponential terms. That way, all possible situations are covered. A.1. Decay Traces wherebyflAem,Aex,t)is a Direrence of Two Exponential Terms. The simulation values of the rate constants kij, ~ Q Iand , k ~ and 2 the calculated values for SI, Sz,and P are
&:,
compiled in Table 1. A value of 0.80 was chosen for whereas E\ was varied from 0.15 to 0.75 in steps of 0.05 to mimic the emission wavelength dependence. Three quencher concentrations [ Q ] were used in the simulations, namely [ Q ] = 0, 0.1, and 0.2 M. For the decay curves at [ Q ] = 0 M, a time increment of 15 ps per channel was used. At [ Q ] = 0.1 and 0.2 M, the timing calibrations were 11 and 9 pdchannel, respectively. The values of the decay times ti as a function of [ Q ] are compiled in Table 2. Since 71 > y2 (TI > z2, koz k ~ z [ Q> ] k o l kzl k ~ l [ Q ]for ) all [ Q ] ,and since 6: = 0.80 > (yl - yz)/(yl - y2 k21), all decays have negative preexponential factors, independent of E:. The computergenerated fluorescencedecay surface had 39 biexponentialdecay traces. In the first series of global compartmental analyses, this fluorescence decay surface was analyzed with IQI kept constant
+ +
+
+
Van Dommelen et al.
8964 J. Phys. Chem., Vol. 99, No. 22, 1995
SCHEME 5
L :
k
TABLE 1: Simulation Values of the Rate Constants kg, ~ Q kQ2 and SI,Sz, and P for the Intramolecular Two-State Excited-State Process Depicted in Scheme 2A kol = 0.4 ns-I k02 = 1 ns-I SI = 0.6ns-' S2 = 1 ns-I kzl = 0.2ns-' k12= 0 ns-I kQI = 1 M-I ns-I P = 0 ns-2 kQ2 = 2 M-I ns-'
I ,
TABLE 2: Values of q as a Function of [Q] Calculated with the Rate Constant Values of Table 1 0 0.1 0.2
1.67
1.oo
1.43 1.25
0.83 0.71
at different values ranging from 0.005 to 0.80 ns-I (Le., scanning according to Scheme 2A). All other fitting parameters were freely adjustable. Unique values of i l and Z.1 will be obtained because sufficient decays with different Z.1 values are included in the ana1y~is.I~ Figure 1A shows Zxiof the global compartmental analyses as a function of bl. It is obvious that Zxiremains practically constant for kol smaller than about 0.5 ns-l. This upper bound was obtained by visual inspection, and it approximates the simulation value of SI. The rate constants k21, k02, and k12 recovered at the various preset values of bl are shown in Figure 1B. This plot indicates that k21 decreases linearly with increasing kol up to a value of about 0.56 ns-I. The estimated values for k02 (XS2) and k12 (X 0) remain virtually constant in the kol range from 0 to 0.5 ns-l. For the parameters SI, S2, and P plotted as a function of kol (Figures 1C,D), plateaus could be observed for bl values smaller than 0.56 ns-'. All these plots evidently indicate that 0 kol < 0.56 ns-'. The visually determined upper bound on kol (0.56 ns-') obtained from Figures 1C,D is in good agreement with the theoretical one (0.60 ns-') calculated from eq 16a using the simulation value of SI (Table 1). The observed plateau values of SI (0.61 & 0.01 ns-I), S2 (1.00 4 0.01 ns-I), and P (0.004 f 0.004 ns-?) were in excellent agreement with the simulation values (Table 1). These values and their corresponding standard deviations were obtained by averaging the values in the plateau region. Figure 1E shows the estimated values of 61as a function of kol. No region of constant 61 values was obtained in the considered kol region. Figure 1F displays the quenching rate constants as a function
of bl. The averaged values in the region of constant SI, S2, and P are ~ Q = I 0.97 f 0.02 M-I ns-I and kQ2 = 2.2 f 0.1 M-' ns-I. Since the scans of bl clearly indicate that P equals zero within experimental error, one can use this information by scanning hl while simultaneously keeping kl2 constant at zero. For this series of global compartmental analyses, Z remained constant x, around -0.97 for bl values smaller than 0.59 ns-'. The values of the rate constant k21 decreased linearly with increasing up to a value of 0.59 ns-I (Figure lG), more in accordance with the simulation value of 0.60 ns-I. The estimated values 2 S2) as a function of bl (Figure 1G) remained constant of ? ~ (= at 0.999 f 0.002 ns-' in the same bl range. Both rate constants showed a smoother profile compared to when k12 was an adjustable parameter (compare Figure 1G to Figure 1B). The same is true for the course of SI(compare Figure 1G to Figure IC),LI (compare Figure 1H to Figure lE), and k ~ and l k~ (compare Figure 11 to Figure 1F) as a function of bl. The average plateau values of SI,~ Q I and , k ~ are z 0.602 f 0.001 ns-l, 0.993 f 0.001 M-' ns-', and 2.24 f 0.01 M-I ns-I, respectively. The plateau value of SI of Figure 1G allows one to obtain limits on bl and k21 which are in excellent agreement with the theoretical ones (eq 16), using the simulation value of SI (Table 1). The value of S2 agrees perfectly with the simulation value of b 2 . In the next series of global compartmental analyses, the same fluorescence decay surface was analyzed with k21 kept constant at different preset values ranging from 0.005 to 0.80 ns-l while all other system parameters were freely adjustable (Le., scanning of k21). The plot of Z as a function of kzl increased x: dramatically for k2l values higher than 0.7 ns-'. The estimated rate constant values as a function of kzl showed that bl decreased linearly with increasing k21 within the range 0.005 to 0.6 ns-I (zSI), whereas in the same region b2 and kl2 remained constant at values of 1.0 ns-' (X S2) and 0 ns-', respectively. The upper bound on k21 obtained from visual inspection agreed very well with the theoretical one (eq 16b) using the simulation value of SI (Table 1). For k21 smaller than 0.6 ns-', the values of SI,S2, and P were constant. From the plots of SI,S2, and P as a function of k21, no lower limit for k21 could be observed. The averaged experimental plateau values of SI (0.61 f 0.01 ns-I), S2 (0.99 f 0.01 ns-I), and P (0.002 f 0.002 ns-*) were in excellent agreement with their corresponding simulation values (Table 1) and with those resulting from scanning bl . In the range where SI,S2, and P are constant, acceptable fits were obtained as judged by Zxi. For the estimated 61 values as a function of k21, no constant values were found in the region of constant SI,S2, and P. The quenching rate constants as a function of k21 yielded plateaus for k21 < 0.61 ns-I. The averaged plateau values for the quenching rate 2 2.18 & constants are kQ1 = 0.97 f 0.02 M-I ns-I and k ~ = 0.02 M-' ns-I. Comparison of the scans of bl and k21 indicates that the same rate constant limits on bl and k21 and the same unique value of k02 are obtained. Therefore, one scan is sufficient to obtain this information. Moreover, the scan of bl with kll kept fixed at zero yields data of better quality. Hence, scanning kol with kl2 held constant at zero is the recommended practice. As Zxris not a good criterion for specifying bounds on the scanned rate constant, in the rest of the paper we will not show anymore figures of Z as a function of the scanned rate constant. 1, Plotting SIand S2 as a function of the scanned rate constant yields indeed more reliable rate constant limits. Let us now investigate whether the model depicted in Scheme
2
J. Phys. Chem., Vol. 99, No. 22, 1995 8965
Distinction between Competing Kinetic Models
2
2
x
9 0
c
f
--F I
? O
--P :g
10
h!
x
0
0
0
o?
" 9r 9r ?r
N r
-
2
x r e
-d xs
x 0
?
0
x
9
0
--F ? -o?
0
0
0
0
2
?
O
0
9
r -
--f
i
xg
x
0
h 0
8966 J. Phys. Chem., Vol. 99, No. 22, 1995 2B can be used to describe the fluorescence decay surface simulated according to Scheme 2A. Therefore, rate constant k02 instead of k~lwas scanned from 0.01 to 0.90 ns-I with k21 kept fixed at zero. All other system parameters were freely adjustable. The parameter estimates are labeled such that the quenching rate constants refer to the same compartments as when kol is scanned. Now it is essential to properly set the constraints on the fitting parameters during the global compartmental analysis. When 61 and 2.1 were restricted to the range [-0.1, 1.11, unacceptable fits were obtained (Zxi > 63) in all cases, independent of the initial parameter guesses. For this reason, no figures are shown as a function of k ~ 2 . When 61 and 2.1 were constrained to the interval [ - O S , 1.51, acceptable fits were obtained as judged by However, the estimated values of 61 and 2.1 were physically unacceptable (outside the range [0, 11). Thus, for a fluorescence decay surface according to Scheme 2A (with negative preexponentials), no altemative solution can be obtained. It must be noted that the distinction between Scheme 2A and 2B can only be made if k2l is held constant at zero during the scan of k02. Using the default constraints on 61 and E l ( [ - O S , 1.51) and treating k21 as a fitting parameter during the scan of l ~ gives 2 acceptable fits with physically acceptable parameter values. When decays with negative preexponentials are included in the fluorescence decay surface with P = 0, it can be concluded that the kinetic model depicted in Scheme 3 (kzl = k12 = 0) is not valid. Indeed, negative preexponential factors can only be obtained when there is an exchange between the excited-state compartments. In this case, Scheme 2 represents the correct kinetic model. For experimental decay traces, standard global (mu1ti)exponential analysis must be used first to determine (i) the number of exponential terms necessary to describe the decay data and (ii) the sign of the recovered preexponentials. If biexponential decays with negative preexponential factors are obtained, repetitive global compartmental analysis as a function of ko, with P set at zero can distinguish reversible from irreversible excited-state processes (see flowchart of Scheme 4). Only one of the two possible Schemes (2A or 2B) can describe the fluorescence decay data adequately. Finally, global compartmental analyses of a fluorescence decay suflace with 6: = 1 and y l > y2 gave the same results whether bl was kept fixed at unity or not. A distinction between Schemes 2A and 2B could be made by scanning hl (respectively ko2) while simultaneously keeping kl2 (respectively k21) at zero. Remember that this fluorescence decay surface has only decay traces whereby f(Aem,Iex,t)is a difference of two exponential terms. A.2. Decay Traces with Only Positive Preexponentials. The second fluorescence decay surface comprised biexponential decay traces with only positive preexponential factors. Now the distinction between Schemes 2 and 3 is no longer evident. The same simulation values as in the previous fluorescence decay surface were used for the rate constants k , , kQ1, and k ~ 2 (Table 1). To produce simulated decays with only positive preexponentials, 6; was set to 0.40 whereas 2.: was varied from 0.30 to 0.75 in steps of 0.05. Three quencher concentrations ([Q] = 0, 0.1, and 0.2 M) were used in the simulations (see Table 2 for the z, values). For the decay curves without added quencher, the time increment per channel was 13 ps, whereas at [Q] = 0.1 and 0.2 M, the timing calibrations were 11 and 9 pskhannel, respectively. Since y~ 72 (TI > z2) for all [QI, and since 6: = 0.40 < ( y l - y2)/(y1 - y2 k21),all decays have positive preexponential factors, independent of the value of Z.;. The computer-generated fluorescence decay surface comprised 30 biexponential decay traces.
Zx,.
+
Van Dommelen et al.
A.2.a. Global Compartmental Analysis. In the first series of global compartmental analyses, the rate constant bl was kept constant at preset values ranging from 0.01 to 0.80 ns-I. The course of the estimated rate constant values as a function of kol is very similar to that depicted in Figure 1B. The rate constant k2, decreases linearly with increasing hl up to a value of about 0.6 ns-I (= SI). The estimated values for h 2 (% SZ)and k12 (z0) remain virtually constant in this range of k~l.For the parameters SI,S2, and P plotted as a function of kol, plateaus could be observed for hl values smaller than 0.58 ns-I. The l averaged plateau values for SI,S2, and P as a function of h are SI= 0.63 f 0.01 ns-', S2 = 0.99 f 0.01 ns-', and P = 0.008 f 0.001 ns-2. These values are in good agreement with the simulation values of Table 1. The upper limit for k-01 obtained from the scanning procedure is in full accord with the upper limit for hl calculated from eq 16a. The parameter bl as a function of bl was not constant in the region where SI,S2, and P were constant. As all the results are very similar to those given in the previous section, with decay traces with negative preexponentials no figures are given. Since P equals zero within experimental error, it is advisable to use this information by scanning kol while simultaneously keeping k12 constant at zero. All other system parameters are freely adjustable. For this series of analyses, Zxiremained constant around -2.28 for hl values smaller than 0.58 ns-'. The values of the rate constant k21 decreased linearly with increasing hl up to a h1value of 0.60 ns-I (Figure 2A). The estimated values of k02 (which equals S2) as a function of bl (Figure 2A) remained constant at 0.994 f 0.003 ns-' for 0 < bl < 0.58 ns-'. Also, SIwas constant (at 0.601 f 0.001 ns-') in the same range of bl (Figure 2A). 61 (Figure 2B) was increasing with higher hl values. The estimated quenching rate l kQ2 (Figure 2c)were constant (at respectively constants k ~ and 1.019 i 0.006 and 2.03 & 0.04 M-' ns-I) in the same QI! region where SI and S2 were invariant. Figure 2A allows one to obtain limits on and k21 which are in excellent agreement with the theoretical ones. Can the model depicted in Scheme 2B be used to describe the fluorescence decay surface simulated according to Scheme 2A (with only positive preexponentials)? To answer this question, b 2 was scanned from 0.01 to 1.20 ns-I while k21 was kept constant at zero during the fittings. All other system parameters were freely adjustable. In contrast to the case with negative preexponentials, acceptable fits (ZY3= -0.86) were obtained for h2 values smaller than 1 ns-'. The rate constants kol and k12 recovered at the various preset values of k02 are shown in Figure 3A. This figure indicates that k12 decreases linearly with increasing k02 up to a b 2 value of 1.00 ns-'. The estimated values of kol (= SI)as a function of ko2 remain constant at 0.60 f 0.02 ns-I for ko2 < 1 ns-'. For S2 plotted as a function of k02 (Figure 3A), a plateau (at 0.958 k 0.006 ns-') could be observed for b 2 values smaller than 1 ns-' . The parameter 61 as a function of h2 is shown in Figure 3B. The l 1.03 f 0.02 M-I estimated quenching rate constants k ~ (= ns-I) and k ~ (= 2 2.28 f 0.04 M-I ns-') were constant in the same I Q ~ region where SI and S2 were invariant. Figure 3A allows one to obtain limits on k02 and k12 which are in excellent agreement with the theoretical ones (eq 17) using the simulation value of S2 (Table 1). The satisfactory fits obtained for the scanning of k02, the smooth profiles of the estimated ~ Q Iand k12 values as a function of h2,and the plateaus observed for SIand S2 all lead to the conclusion that, for a fluorescence decay surface comprising decay traces with only positive preexponentials, a nontrivial k ~ land ) altemative solution with the same combinations of (SI,
J. Phys. Chem., Vol. 99, No. 22, 1995 8967
Distinction between Competing Kinetic Models 1.2
1
1
0.8 0.8 0.6
0.6 0.4
0.2
0 0
0.1
4
0
0.2
0.4
0.0
0
0.2
0.4
0.6
0.8
1
1.2
0.2
0.4
0.6
0
0.2
0.4
0.6
0.8
1
1 2
kO1 (
2.4
4
-
2.2 -
c
ko2( (ns) -l)
-')
al
Figure 2. Global compartmental analyses of the fluorescence decay surface (Table 1) with only positive preexponentials. Scanning as a function of kol with k l ~kept fixed at zero duringjhe analyses. (A) Estimated values of kzl, ko2 (= Sz), and SI. (B) bl. (C) Estimated quenching rate constant values kal and k ~ 2 .
(S2, kQ2) can be found, depending on which rate constant is being scanned (bl or b 2 ) . In this case, no distinction can be made between the two nontrivial altemative solutions (Table 3). Note that interchanging the indexes 1 and 2 in Table 3 gives the trivial altemative solutions, corresponding to switching the labels of the compartments. Decay traces with positive preexponentials and P = 0 can be described by either Scheme 2 or 3. Is it possible to distinguish between these two altemative kinetic models by the Zxi goodness-of-fit criterion? In order to answer this question two different compartmental analyses were performed using linking Scheme 5. In the analysis according to the correct kinetic model (Scheme 2A), the rate constants bl and k12were kept constant at their simulation values of 0.40 and 0 ns-', respectively (Table 1). In this case, a good fit was obtained
Figure 3. Global compartmental analyses of the fluorescence decay surface (Table 1) with only positive preexponentials. Scanning as a function of k02 with kzl kept fixed at zero duriFg the analyses. (A) Estimated values of k12, kol (= SI),and SZ.(B) bl.
(Zx: = -1.50). The estimated values that were obtained from this compartmental analysis are given in Table 4A. The second analysis with both rate constants k21 and k12 fixed at zero = (incorrect kinetic model, Scheme 3) also gave a good fit (Zx: -0.44). The estimated parameters of this compartmental analysis are compiled in Table 4B. No distinction can be made even when between Schemes 2 and 3 on the basis of Zx,, simulationvalues of some rate constants are used in the analysis. Table 4A shows that, if the proper kinetic model is applied, the parameter estimates are accurate and precise (compare to Table 1). Applying the wrong kinetic model (Table 4B) results in inaccurate parameter estimates although an excellent fit is obtained. The estimate for bl equals SI,due to the fact that k21 = 0; the numerical estimates for kQ2 and b, are too high. From these results, it can be concluded that, for a system with P = 0 (known from scanning) and only positive preexponentials (known from global biexponential analysis), it is impossible to distinguish between the two different kinetic models (Schemes 2 and 3) on the basis of the goodness-of-fit criterion Z alone. x: The question remains: is there a method to distinguish between these two different kinetic models? This will be investigated in the next section. A.2.b. Comparison of DAS and SAEMS. We will examine now how to distinguishbetween the two different kinetic models depicted in Schemes 2 and 3 if P = 0 and if only decay traces with positive preexponentials are enclosed in the fluorescence decay surface. Furthermore, we will also investigate whether it is possible to make a distinction between the two altemative solutions (corresponding to Schemes 2A and 2B) on the basis of the SAEMS and DAS.
8968 J. Phys. Chem., Vol. 99, No. 22, 1995
Van Dommelen et al.
TABLE 3: Limits on the Rate Constants k i and the Estimated Values of kQl, kQ2,SI, and S2 (A) Scanning of kol with k12 = 0 (Scheme 2A) kQl = 1.019 i 0.006 M-lns-' kp? = 2.03 ik 0.04 M-'ns-'
0 < kol < 0.601 ns-' 0 < kzl < 0.601 ns-l kol = 0.994 ns-I
(B) Scanning of koz with k21 = 0 (Scheme 2B) kul = 1.03 i 0.02 M-'ns-' kQ2 = 2.28 ik 0.04 M-lns-'
kol = 0.60 ns-I 0 < k02 < 0.958 ns-I 0 < k12 < 0.958 ns-'
TABLE 4: Parameter Values Estimated by Global Compartmental Analysis. The Simulation Values of the Rate Constants are Compiled in Table 1 and 6: = 0.40
i.:
SI
61
koi rns-ll k?i l ~ ' 1 kQl [~-lnS-']
ko2 r ~ ' 1 kiz 1 n 5 - ' 1 k ~ [hl-'rl-'] z
A: koj and klz Kept Constant at Simulation Values of Table 1 (Zyz= -1.50) 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
0.32 i 0.04 0.37 i 0.04 0.42 i 0.04 0.47 i 0.04 0.52 i 0.03 0.57 f 0.03 0.62 i 0.03 0.67 i 0.03 0.72 i 0.02 0.76 i 0.02
0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
0.35 i 0.06 0.38 0.06 0.41 i 0.06 0.45 i 0.06 0.49 i 0.06 0.52 i 0.05 0.56 i 0.05 0.60 i 0.05 0.65 i 0.04 0.69 i 0.04
0.39 i 0.01
0.4 (c) 0.20 i 0.01 1.01 i 0.01
B: kll and kll Kept Constant at Zero (Zzi = -0.44) h 0.6 i 0.2 0.593 f 0.001 0.95 0 (c) 0 (c) 1.071
* 0.005
*
1 .oo 0.01 0 (c) 2.00 f 0.04
* 0.01
2.38 i 0.01
One may expect that a distinction between the two kinetic models depicted in Schemes 2 and 3 can be made by comparing DAS and SAEMS. SAEMS can be constructed in two different , 61, ?I} estimates ways. (i) By using [Q] and {k,,, k ~ l kQ2, obtained from a scan of kol (respectively, ko2) with kl2 (respectively, k21) kept fixed at zero. Only those parameter estimates from the range where SI and S2 are constant (the plateau region) may be used. The so-constructed SAEMS will be calledplateau SAEMS. (ii) By using [Q]and the parameter values estimated by a single global compartmental analysis where k21 and klz are both kept constant at zero. These SAEMS will be called zero SAEMS. If k21 and k12 are not both equal to zero, the DAS will differ from the pluteau SAEMS as well as from the zero SAEMS. In that case, the zero SAEMS are also distinct from the plateau SAEMS. In the case that k21 and k12 are both equal to zero, the DAS, the plateau SAEMS, and the zero SAEMS will all coincide. Let us examine if that is indeed the case for the system simulated according to Scheme 2A (Table 1). Using a synthetic fluorescence spectrum, the true DAS at [Q] = 0 M (Figure 4A) were computed (eq 19) using the simulated preexponentials and decay times (Table 2). The true SAEMS (eq 18) at [Q] = 0 M computed with the simulation values of the rate constants ko (Table l), 6:, and are displayed in Figure 4B. The emission wavelengths 1'" varied linearly with the values. It is evident that the true DAS do not coincide with the true SAEMS. The constructedplateau SAEMS (fitting according to Scheme 2A with ki2 fixed at zero) at kol = 0.01
e',
SI= 0.601 i 0.001 ns-' S? = 0.994 i 0.003 ns-'
SI = 0.60 i 0.02 ns-I
SZ= 0.958 i 0.006 ns-l
ns-', at kol = 0.40 ns-I (simulation value), and at kol = 0.56 ns-I are a function of k01. The plateau SAEMS at kol = 0.40 ns-I coincides very well with the true SAEMS, whereas the other two plateau SAEMS at the boundaries of the range of constant SI and S2 differ from the true SAEMS (figures not shown). In this case, the constructed plateau SAEMS are not unique, and only limits on the SAEMS can be obtained. The plateau SAEMS (fitting according to Scheme 2B with k?l fixed at zero: nontrivial alternative solution) at k02 = 0.01 ns-l and at ko2 = 1.00 ns-' are completely different from the true SAEMS. The incorrect kinetic model results in inaccurate SAEMS, although excellent fits are obtained in the range of constant SI and Sr values. Since it is easier to compare S2, values (eq 18b) than SAEMS (eq 18a), graphs of Q l are shown in Figure 4C. These plots indicate that only the Q l values at kol = 0.40 ns-I with kl2 = 0 ns-I coincide very well with the true '2,. The zero SAEMS are shown in Figure 4D. They are significantly different from the true SAEMS and slightly different from the true DAS, indicating that the kinetic model of Scheme 3 (k2l = k12 = 0) can be excluded. A.3. Decay Traces whereby f(l.'*,+,t) is a Sum or Difference of Two Exponential Terms. The simulation values of the rate , k ~ and 2 the calculated values for SI, S2, constants k,,, ~ Q I and and P are compiled in Table 5. A value of 0.90 was chosen for 6: whereas 2: had values of 0.65, 0.60, 0.55, 0.53, 0.50, 0.47,0.45,0.43,0.40,and 0.35. Three quencher concentrations [Q]were used in the simulations, namely [Q]= 0.1, 0.2, and 0.3 M. For the decay traces at [Q]= 0.1 M, a time increment of 13 ps per channel was used. At [Q]= 0.2 and 0.3 M, the timing calibrations were 10 and 5 pskhannel, respectively. The values of the decay times z, as a function of [Q]are compiled in Table 6. For the considered system, we have y l < y~ (tl < t?,ko2 k ~ z [ Q
