J. Phys. Chem. 1996, 100, 4879-4887
4879
Identifiability of Irreversible Intermolecular Two-State Excited-State Processes Noe1 l Boens,* Andrzej Kowalczyk,† and Els Cielen Department of Chemistry, Katholieke UniVersiteit LeuVen, B-3001 HeVerlee, Belgium ReceiVed: NoVember 6, 1995X
This report analyzes which kinetic and spectral parameters can be recovered by global compartmental analysis for intermolecular two-state excited-state processes when k21[M] (Scheme 1) is negligible. The time-resolved fluorescence of such a photophysical system will decay as a biexponential function with invariant decay times. The ratio of the amplitudes associated with the decay times will depend on [M]. Although the association rate constant k21 is known to be zero, this knowledge is not helpful in determining unique values for the remaining parameters. It is shown that the available information which can be derived from timeresolved fluorescence data only is strongly limited. When k12 is not negligible, the values of k01, (k02 + k12), and the relation connecting k12 and the ground-state dissociation constant can be obtained from two decay traces collected for two different concentrations of M. For such a system it is impossible to determine the ground-state dissociation constant from time-resolved fluorescence measurements exclusively. The identifiability analysis shows that additional decay curves measured at different concentrations of M and excitation or emission wavelengths do not provide independent information useful for the unique determination of the remaining parameters. If the ground-state dissociation constant can be obtained from fluorimetric titration, the photophysical system becomes identifiable. In the limiting case when k12 ) 0, the relation between k12 and the ground-state dissociation constant vanishes. The addition of quencher to intermolecular two-state excited-state processes when k21[M] is negligible does not provide additional information useful for the unique determination of the kinetic and spectral parameters.
1. Introduction
SCHEME 1 fluorimetry1,2
Time-resolved has proven to be an invaluable tool for studying the molecular dynamic behavior of excitedstate species on subpicosecond to microsecond time scales. To identify the kinetic model for the excited-state processes, a fluorescence decay surface is created by collecting decay traces measured under different conditions (such as pH, temperature, excitation and emission wavelengths, concentration of coreactant, added quencher, etc.). Compared to individual curve analysis, simultaneous (or global) analysis3-5 of the decay surface results in a more accurate parameter estimation and a better discrimination between competing models. Global compartmental analysis6-9 allows the direct determination of the rate constants of excited-state processes and the spectra associated with excited-state species. This method describes a photophysical system in the most direct way by taking into account the relations that may exist between the parameters of interest. The global compartmental analysis approach has been applied to a variety of experimental intermolecular two-state excited-state processes.10-15 In this paper we describe the kinetics and identifiability of intermolecular two-state excited-state processes whereby the association rate is negligible, resulting in invariant decay times. It is, however, assumed that the amplitudes associated with the decay times depend on the coreactant concentration. The conclusions of the theory are tested by global compartmental analyses of computer-generated fluorescence decay surfaces. 2. Theory 2.1. Fluorescence Decay Kinetics. Consider a casual, linear, time-invariant, intermolecular system consisting of two * To whom correspondence should be addressed. † On leave from the Nicholas Copernicus University, 87-100 Torun, Poland. X Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-4879$12.00/0
distinct types of ground-state species and two corresponding excited-state species as depicted in Scheme 1. Ground-state species 1 can reversibly react with coreactant M to form groundstate species 2. Excitation by light creates the excited-state species 1* and 2*, which can decay by fluorescence (F) and nonradiative (NR) processes. The composite rate constants for these processes are denoted by k01 ()kF1 + kNR1) and k02 ()kF2 + kNR2). k21 denotes the second-order rate constant for the association 1* + M f 2*, while k12 is the first-order rate constant for dissociation of 2* into 1* and M. In this paper we will consider only the case where k21[M] is negligible. If the system shown in Scheme 1 is excited with a δ-pulse which does not significantly alter the concentrations of the ground-state species (i.e., in the low excitation limit), the fluorescence δ-response function, f(λem,λex,t), at emission wavelength λem due to excitation at λex is given by7
f(λem,λex,t) ) κ c˜ (λem)U exp(tΓ) U-1b˜ (λex), t g 0 (1) with κ ) ∑∀ibi∑∀ici a proportionality constant. U ≡ [U1, U2] is the matrix of the two eigenvectors of the compartmental matrix A (eq 2), and U-1 is the inverse of U. γ1 and γ2 are the eigenvalues of A corresponding to U1 and U2, respectively, and exp(tΓ) ≡ diag{exp(γ1t), exp(γ2t)}.
A)
[
-k01 k12 -(k02 + k12) 0
]
b˜ (λex) is the 2 × 1 vector with elements b˜ i(λex) defined by © 1996 American Chemical Society
(2)
4880 J. Phys. Chem., Vol. 100, No. 12, 1996
Boens et al.
b˜ i ) bi/(b1 + b2)
(3)
where bi denotes the concentration of i* at time zero:
bi ) [i*]t)0
(4)
which, in the low excitation limit, is proportional to the groundstate absorbance of i. Hence, in the low excitation limit (as in single-photon timing experiments1,2), b˜ i represents the normalized absorbance of species i at λex. The elements bi (and b˜ i) depend on [M]. c˜ (λem) is the 1 × 2 vector of the normalized emission weighting factors c˜ i(λem) of species i* at λem:
c˜ i ) ci/(c1 + c2)
(5)
The emission weighting factors ci(λem) are given by
ci(λem) ) kFi∫∆λem Fi(λem) dλem
(6)
kFi represents the fluorescence rate constant of species i*; Fi(λem) is the emission density of species i* at emission wavelength λem, normalized to the complete steady-state fluorescence spectrum Fi of species i*; ∆λem is the emission wavelength interval around λem where the fluorescence signal is monitored. Fi(λem) is defined by
Fi(λ ) ) Fi(λ )/∫full bandFi dλ em
em
em
(7)
the ground-state dissociation constant. When [M] is below the concentration corresponding to numerical value of the groundstate dissociation constant Kd, the absolute value of the amplitude associated with the limiting value of τ2 ) (k02 + k12)-1 ) S2-1 decreases to zero as [M] f 0. Simultaneously the amplitude associated with the limiting value of τ1 ) k01-1 asymptotically approaches a value proportional to b1c1. This results in a monoexponential decay with τ ) k01-1 when [M] f 0, and assigns unique values to k01 and S2 (see below: Identifiability). 2.2. Identifiability. In this section we will discuss whether the parameters k01, k02, k12, b˜ , and c˜ can uniquely be determined by algebraic manipulations of γi and Ri, the descriptive parameters of f(λem,λex,t) obtained at different known concentrations of the coreactant M, and at various excitation and emission wavelengths. Therefore, the aim of the identifiability study is to investigate whether it is possible to derive expressions for the parameters k01, k02, k12, b˜ , and c˜ as a function of γi and Ri (which are assumed to be exactly known). Unfortunately, the equations relating Ri (eq 11) to kij, b˜ , and c˜ are nonlinear, and, therefore, the original set of equations is replaced by equivalent (simpler) ones. The first set of equations is generated by the elementary symmetric functions σi (i ) 1, 2) in γ1 and γ2 defined by16
σ1 ≡ γ1 + γ2
(12)
σ2 ≡ γ1γ2
(13)
Equation 1 can be written in the common biexponential format:
f(λem,λex,t) ) R1 exp(γ1t) + R2 exp(γ2t), t g 0
(8)
The exponential factors γ1,2 are given by
γ2 ) -(k02 + k12) ) -S2
(9b)
Since σ1 and σ2 are symmetric functions in k01 and S2, eqs 14 and 15 have interchangeable solutions for k01 and S2. Therefore, on the basis of the decay times of a single decay trace, there are two possibilities to assign numerical values to k01 and S2. However, as mentioned before, the preexponentials as a function of [M] can resolve this ambiguity. For [M] f 0, only the preexponential factor associated with k01 remains and f(t) becomes monoexponential:
(10)
] [
(1 - b˜ 1)k12 γ2 - γ1
) κc˜ 1 (1 - b˜ 2) -
b˜ 2k12
]
γ2 - γ1
(11a)
(
(15)
σ2 ) k01 (k02 + k12) ) k01S2
The exponential factors γ1,2 (and hence the decay times τ1,2) depend exclusively on the rate constants k01, k02, and k12, but are independent of k21[M]. The preexponential factors R1,2 are dependent on k01, k02, k12, b˜ (λex), and c˜ (λex), as is evident from eq 11.
R2 )
(14)
(9a)
γ1,2 ) -1/τ1,2
[
σ1 ) -k01 - k02 - k12 ) -k01 - S2
γ1 ) -k01
and are related to the decay times τ1,2 according to
R1 ) κc˜ 1 b˜ 1 -
The functions σ1 and σ2 are functions of kij only. The explicit expressions corresponding to eqs 12 and 13 are given by
κ(1 - b˜ 1) 1 - c˜ 1 +
c˜ 1k12
) (
γ2 - γ1
) κb˜ 2 1 - c˜ 1 +
c˜ 1k12 γ2 - γ1
)
(11b) The preexponentials R1,2 can still be dependent on [M] because of the ground-state dependence of b˜ on [M]. The expressions of the amplitudes R1,2 are more complex than those of τ1,2 because R1,2 depend on both excited-state (kij and c˜ ) and groundstate (b˜ ) parameters. A change of [M] will influence the groundstate composition according to the ground-state equilibrium. That will be reflected in the preexponentials: they will change in the concentration range of M close to the numerical value of
f(λem,λex,t) ) b1c1 exp(-k01t), t g 0
(16)
Consequently, if a decay trace at very low [M] can be obtained, unique numerical values can be assigned to k01 and S2. It is, however, never possible to assign unique values to the individual rate constants k02 and k12. Only an upper limit equal to S2 can be specified for both k02 and k12 (the lower limit is zero).
0 < k02 < S2
(17a)
0 < k12 < S2
(17b)
The second set of equations is provided by the Markov parameters of the system which are defined by16
mi ≡ f(i)(0) ≡ {∂(i)f(t)/∂ti}(0), i ) 0, 1
(18)
with f(i)(0) the ith time derivative of the fluorescence δ-response function f(t) at time zero. For a dual exponential f(t) (eq 8), the Markov parameters are
Intermolecular Two-State Excited-State Processes
J. Phys. Chem., Vol. 100, No. 12, 1996 4881
2
mi ) ∑Rjγji, i ) 0, 1
(19)
j)1
with γji the ith power of γj. The Markov parameters mi can also be expressed as a function of the rate constants and the spectral parameters,
mi ) κc˜ Aib˜ , i ) 0, 1
(20)
The specific expressions for m0 and m1 are
m0 ) κ(b˜ 1c˜ 1 + b˜ 2c˜ 2) m1 ) κ{(-b˜ 1k01 + b˜ 2k12)c˜ 1 - [b˜ 2(k02 + k12)]c˜ 2}
(21a) (21b)
m0 and m1 depend on [M] only through b˜ 1 and b˜ 2 (i.e., through the ground-state equilibrium). Furthermore, m0 and m1 are also dependent on the excitation and emission wavelengths. For a decay trace at [M] collected at emission wavelength λem due to excitation at λex, substitution of eqs 3 and 5 in eq 21 and elimination of the scaling factor κ from m0 and m1 yields eq 22:
-b˜ 2[(m0k12 + P - Q)c˜ 1 + Q] + Pc˜ 1 ) 0
(22)
P ) m1 + m0k01
(23)
Q ) -m1 - m0(k02 + k12) ) -m1 - m0S2
(24)
with
P and Q generally depend on [M] and on both the excitation and emission wavelengths. Note that eq 22 is a homogeneous nonlinear equation with three unknowns, namely k12, b˜ 2, and c˜ 1. Equation 22 has a trivial solution for b˜ 2 ) 0 (b˜ 1 ) 1), c˜ 1 ) 0, and k12 any value. If the trivial solution is excluded, dividing eq 22 by b˜ 2c˜ 1 converts it into a linear homogeneous equation with three unknowns, k12, (1 - 1/b˜ 2), and (1 - 1/c˜ 1):
m0k12 + P(1 - 1/b˜ 2) - Q(1 - 1/c˜ 1) ) 0
(25)
For a single decay trace collected at one concentration of M and at one emission wavelength λem due to excitation at λex, eq 25 provides the basic equation from which values of the rate constant k12 and the spectral parameters b˜ 2 and c˜ 1 must be derived. Now we will investigate if we can reduce the required a priori information by collecting decay traces under different experimental conditions such as different [M], λem or λex. By constructing a similar expressions at the same emission wavelength, but at a different concentration of M (corresponding to a different b˜ 2), c˜ 1 can be eliminated. The resulting expression contains in addition to k12, the unknown elements b˜ 2k and b˜ 2l at [M]k and [M]l, respectively:
(Qkm0l - Qlm0k)k12 + PkQl(1/b˜ 2k - 1) PlQk(1/b˜ 2l - 1) ) 0 (26) where the subscripts k and l refer to the concentrations [M]k and [M]l. After substituting P and Q (eqs 23 and 24) in eq 26 one has
(m1lm0k - m1km0l)(k12 - S2 + k01) + PkQl(1/b˜ 2k) PlQk(1/b˜ 2l) ) 0 (27) By choosing extra emision wavelengths with different c˜ 1 one can construct additional equations of the type of eq 27. However, selecting another emission wavelength provides no linearly independent equations. Indeed, c˜ 1[k01 - S2 + c˜ 1(S2 -
k01 + k12)] is a common factor in the coefficients (m1lm0k m1km0l), PlQk, and PkQl. Therefore, choosing a different emission wavelength changes all coefficients in the same proportion. To have an identifiable photophysical system, two out of the three unknowns (k12, b˜ 2k, and b˜ 2l) have to be known beforehand. It must be emphasized that c˜ 1 can in principle also be eliminated from eq 25 by selecting another excitation wavelength with a different b˜ 2 value. An alternative strategy consists of eliminating b˜ 2 from eq 25. and From two decay traces at two emission wavelengths λem i , due to excitation at the common excitation wavelength λex λem j and concentration [M], we can construct two eqs 25 with identical b˜ 2. Elimination of b˜ 2 from these equations leads to
(m1im0j - m1jm0i)(k12 + S2 - k01) + PiQj(1/c˜ 1j) PjQi(1/c˜ 1i) ) 0 (28) where the subscripts i and j refer to the emission wavelengths and λem λem i j . Additional equations of the type of eq 28 at different excitation wavelengths (or [M]) can be built. Choosing another emission wavelength, however, yields no linearly independent equations. Indeed, b˜ 2[k01 - S2 + b˜ 2(S2 - k01 k12)] can be factored out from the coefficients (m1im0j - m1jm0i), PiQj, and PjQi. Therefore, selecting a different excitation wavelength changes all coefficients proportionally. To have an identifiable photophysical system two out of the three unknowns (k12, c˜ 1i, and c˜ 1j) have to be known a priori. Note that there are two interchangeable numerical solutions for S2 and k01 (see identifiability using σ1,2). Each of the solutions leads to separate eqs 25-28 describing different relationships between the rate constants and the spectral parameters. The spectral parameters b˜ 2 can be expressed as a function of the ground-state dissociation constant Kd, the molar extinction coefficients i(λex) at λex of ground-state species i, and the known coreactant concentration [M]. For b˜ 2k at [M]k we have
1(λex)Kd 1 )1+ b˜ 2k 2(λex)[M]k
(29)
By constructing a similar expression at a different concentration of M, say [M]l, eqs 26 and 27 can be rewritten (eq 30) as a function of [M]k, [M]l, and Kd1/2 instead of b˜ 2k and b˜ 2l.
(m1lm0k - m1km0l)k12 +
(
PkQl
[M]k
-
)( )
PlQk 1Kd ) 0 (30) [M]l 2
Equation 30 is a homogeneous equation in two unknowns, k12 and Kd1/2. Using a third [M], say [M]m, provides a second equation of the type of eq 30. It can be shown that this homogeneous system is linearly dependent. Indeed, the coefficients of k12 and Kd1/2 have as common factor ([M]l - [M]k)/ [(1Kd/2 + [M]k)(1Kd/2 + [M]l)] and depend on [M]k and [M]l in the same way. Therefore, the eq 30 obtained at additional concentrations of M will only differ by a multiplication factor. Consequently, the value of Kd1/2 can be expressed as a linear function of k12. In conclusion, from two decay traces for two different [M] at λem due to excitation at λex one can obtain a value of k12 provided Kd1/2 is known. This value may be obtained independently from ratiometric fluorescence titration, as is explained below. If Beer’s law is obeyed and if the absorbance at the excitation wavelength λex is low (
