J . Phys. Chem. 1993,97, 799-808
799
ARTICLES Kinetics and Identifiability of Intramolecular Two-State Excited-State Processes with Added Quencher. Global Compartmental Analysis of the Fluorescence Decay Surface No61 Boens,'*?Marcel Ameloot,* Bart Hermans,? Frans C. De Schryver,? and Ronn Andriessent Department of Chemistry, Katholieke Universiteit Leuven. 8-3001 Heverlee, Belgium, and Limburgs Universitair Centrum, B-3590 Diepenbeek. Belgium Received: September 18, 1992; In Final Form: October 30, 1992
The fluorescence decay analysis of intramolecular two-state excited-state processes with added quencher is discussed in terms of compartments. The kinetics specifying the two excited-state species concentrations are derived. The fluorescence decay surface is expressed in terms of the system parameters, namely the rate constants and the spectroscopic parameters 81 and E l . 61and T.1 are respectively the relative absorbance and the normalized spectral emission weighting factor of species 1. The report investigates the prerequisites for obtaining the unique set of system parameters. The results of this identifiability study indicate that the following conditions have to be satisfied in order to make an intramolecular two-state excited-state system with added quencher identifiable. First, the fluorescence decay surface must include at least one set of decay traces measured for a minimum of three different quencher concentrations a t the same excitation/emission wavelength setting. One of the quencher concentrations used may be equal to zero. Second, the rate constants of quenching of the two excited species must be different. Third, at least one system parameter, which is not a rate constant of quenching, must be known. Under these conditions, four sets of system parameters are mathematically possible. If the known system parameter is a rate constant, decay traces of a suitable model compound measured at a minimum of two quencher concentrations must be included in the analysis in order to obtain the unique set of values for the rate constants. The unique set of ( 6 1 , E l ) values can be recovered by including decay curves at a minimum of two quencher concentrations and at an additional excitation wavelength with a different or at another emission wavelength with a different El. If the known system parameter is a 81value different from zero and unity, the fluorescence decay surface must include at least nine decay traces measured at four emission wavelengths with different E1 (corresponding to at least three quencher concentrations at the first emission wavelength and at least two quencher concentrations at the other three emission wavelengths), to uniquely determine the set of system parameters. If the known system parameter is a El value different from zero and one, a t least four excitation wavelengths with different 61 are necessary to obtain the unique set of system parameters. The conclusions of the identifiability study are confirmed by the results from the global compartmental analysis of computer-generated fluorescence decay traces.
SCHEME I
1. Introduction
The power and performance of the simultaneousanalysis1v2of fluorescence decay traces have been widely demonstrated. To benefit the most from the merits of this global analysis approach, one has to fit for the rate constants and species-associated spectra.3-5 In a recent paper6we discussed the global compartmental analysisof the fluoresctncedecay surfaceof intramolecular two-state excited-state processes without added quencher. The kinetics specifying the two excited-state species concentrations were derived. The fluorescence decay surface was expressed in terms of the rate constants and the spectroscopic parameters 61 and ZI.61and El are respectively the relative absorbance and the normalized spectral emission weighting factor of species 1. It was shown that at least three parameters of the compartmental system must be known beforehand to obtain all relevant kinetic and spectroscopicinformation about the intramolecularprocesses. Unfortunately, it is nearly impossible to get reliable values of three system parameters. As a result of the assumptions that have to be made about some parameter values, the results obtained by this method are less reliable. Therefore, it is important to develop a methodology with a minimal number of assumptions.
*
+
To whom correspondence should be addressed. Katholieke Universiteit Leuven. Limburgs Universitair Centrum.
0022-3654J58J2097-0799$04.00/0
F I2& r;-1 kQl[Q'
1
k21
1'01
kQ2R
hV
1 1 kO2
hV
FI=r;l In this paper we shall formulate the fluorescence decay analysis of intramolecular two-state excited-state processes with added quencher in terms of compartments. First, the kinetics describing the excited-state species concentrations will be derived and expressions for the fluorescence decay surface will be given. Second,the structural identifiability will be investigated. Third, the estimation of the unknown parameters from computergenerated fluorescence decay traces will be reported. 2. Kinetics a. Rate Equatiolg for Excited-StateSpecies. Consider a causal, linear, time-invariant, concentration-independent (intramolecQ 1993 American Chemical Society
Boens et al.
800 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
ular) system consisting of twodifferent excited-state species with added quencher. The kinetic model for such an intramolecular bicompartmental system with added quencher is depicted in Scheme I. Ground-state species 1 can undergo a reversible transformation to 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 for those processes are denoted by k01 ( = ~ F I + klcl + k1scl) and k02 ('kF2 + k 1 ~ 2 + klsc2). The rate constant describingthe transformation 1* 2* is represented by k2l and for the transformation 2* I * by k12.The additionof a quencher Q to the bicompartmentalsystem acceleratesthe depopulationof the excited states by ~ Q[Q] I [ 1*I and ~ Q ~ [[2*] Q ] for species 1* and 2*, respectively. It is assumed that the added quencher Q affects only the excited-state species deactivation and does not change in any way the ground-state equilibrium. Note that all rate constants are nonnegative. If the system depicted in Scheme I is excited by a &pulse, the concentration of the two excited species 1* and 2* as a function of time is described by the following system of differential equations
--
(1) or (6) is given by7 X*(t) = exp(tA)b with exp(rA) defined as (tA)"
+ E- n! OD
exp(tA) = I
(9)
where I is the identity matrix. Usually it is preferableto compute exp(tA) by using eigenvalues rather than attempt to sum the series (10). Assume that the 2 X 2 compartmental matrix A has two linearly independent eigenvectors P I and P2 associated with the eigenvalues y l and 7 2 , respectively, Le., A = P l F with P = [ P I ,P2] and P I the inverse of the matrix of the eigenvectors, and 'I = diag(yl, y2). Equation 9 can then be written as
X*(t) = P exp(tr)P-'b (11) where exp(tr) = diag(exp(ylt), exp(y2r)). P and exp(rr) are functionsof the rate constants kol,k21rk ~k02,~ kI2, , and kQ2,and the quencher concentration [Q]. b is dependenton the excitation wavelength P.Equation 1 1 can explicitly be expressed as a sum of two exponentials [1*1(0 = 811 exp(7,t) + 812 exp(72t)
(124
with the concentrations at time zero given by
Defining the matrices A, X*(r), and X*'(t) for the bicompartmental system
-(hi + k21 + ~ Q I [ Q ] )k12 A = [k2,
]
-(k~2+ k12 + ~ Q ~ [ Q I )
(3)
(4)
The preexponential factors are
we can write the differential equations (1) in matrix notation as
X*'(t) = AX*(t), r 1 0 and the initial conditions (2) as
(6)
X*(O) = b (7) The 2 X 2 time-independent 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 the jth column, is -(ko, + kw[Q]), the negative sum of the deactivation processes of thejth compartment in the absenceof transformation processes between theexcited-statecompartments. The elements bi of the 2 X 1 column vector b are related to the ground-state absorbances of species i by
822
(164
If k21 = 0 the time dependence of the concentrations of excited species 1' and 2* is given by [l*I(t) = Ol1 exp(7,t) + BI2 exp(7,t)
with
and where cI and c2 are the molar extinction coefficients of 1 and 2, respectively. The solution of the system of differential equations
= ib2W2 + 71) - bIk,ll/(72 - 71)
U7a)
Intramolecular Two-State Excited-State Processes
The Journal of Physical Chemistry, Vol. 97, No.4, 1993 801
The decay of [2*] is monoexponential while that of [1*] is biexponential. In the complementarycase, when k12= 0, the timedependence of the concentrationsof excited species 1 * and 2* is given by eqs 17-19 where the numbers 1 and 2 are interchanged. If k12 = 0 and k21 = 0, the concentrations of both excited species 1 and 2* decrease monoexponentially,
compartmental analysis, one fits directly for the rate constants kol, k21, k ~ l ,k02, kl2, and k ~ 2 ,the normalized zero-time concentrations61, and the normalized spectral emission weighting factors E l . If the compartmental matrix A has two distinct eigenvalues y1 and 72 with corresponding eigenvectors P I and P2, respectively, eq 29 can be formulated as f(Xcm,Xcx,t) = KE(X'~)P exp(tI") P-'&(Xex)
with
b. Fluorescence Decay Kinetics. In fluorescence decay experiments,onedoesnot observeX*(r) directly, but the composite spectral emission contoursof the excited-state species. Therefore, the fluorescence 6-response function, flXCm,XCX,t), measured at emission wavelength XCmand due to excitation at Xcxis expressed bY
= c X * ( t ) = c(Xcm)exp(tA) b(Xex) (22) where c is the 1 X 2 vector [CIcq] of spectral emission weighting f(Xcm,Xcx,t)
factors ci(Xcm)given by
ci(Acm)= k,,
pi(Xcm) dX"
JAAem
kFi is the fluorescence rate constant of species i; pi(Xcm)is the spectral emission density of species i at emission wavelength XCm, normalized to the complete emission band, and AXcmis the emission wavelength interval in which the fluorescenceis monitored. pi(Xcm) is defined by
(30)
3. Structural Identifhbility of the Iatrrmokculrr Bicompartmental System with Added Quencber a. Identifiability Equations. In this section we shall examine the conditions under which the system parameters kol, k21,~ Q I , ko2, k12, kQz, 61,and E l can be uniquely determined from the measurementof the fluorescence decay surface of intramolecular two-state excited-state proctsses with added quencher (seeScheme 1). For the case considered in this paper the fluorescence decay trace d,(Xcm,Xcx,t) measured at XCm and due to excitation at Xcx is the convolution of flt) and the instrument response function u(Xcm,XcX,t)*
d,(t) = K ' E exp[(t-t?A] ~ bu(t? dt'
(31)
in which K' is a scaling factor which includes K . From eq 31 it is clear that the model parameters must be identified through f l t ) . The problem of structural identification of the model can now be described as follows: is it possible to uniquely determine the unknown parameters kol, k21, kpl, k02, k12, k ~ 261, , and E l from knowledge of the fluorescence &response function At)? 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 4. The following formulas relate the parametersof the fluorescence &response functionf l t ) to the unknown system parameters kol, k21, ~ Q Ik02, , kl2, k ~ 261, , and E l . First, information is available from the characteristic polynomial of A which can be expressed as9 2
whereFddenotesthesteady-statefluorescencespectrumofspecies i*. If the 2 X 2 matrix A has two distinct eigenvalues y1 and 7 2 , eq 22 can be expressed as f(Xem,Xcx,t) = I C l B l l
+ c28211 exp(y1t) + [c1812 + c28221 exp(72t) ( 2 5 )
det(y1 -A) = n ( y - y i ) = y2- u,y
+ u2
(32)
i= 1
where the ui are the elementary symmetric functions in yI,7z9 and are given by
(33)
which can formally be represented as f(Xem,Xex,t) = a1exp(y,t)
+ a2exp(y2t)
(26)
with ai the ith preexponential. If the elements bi of b are normalized as
(34) From q s 14 and 15 at quencher concentration [Q]kthe following expressions of uI and u2 are derived:
and similarly the elements ci of c as
E, = C i / C C j i
eq 22 can be written as .f(XCm,XCX,t) = K E ( X ' ~ )exp(tA) &(AeX)
(29) with K a proportionality constant. The use of K , 6i, and E; allows one to link 6j and Eiin the data analysisso that the collected decay traces are not required to be scaled. Indeed, b(XCX) depends just on the excitation wavelength whereas E(XCm) depends on the emission wavelength only. In our implementation of global
withpi)(0) the ith time derivativeof the fluorescence &response functionflt) at time zero. Ifflr) is biexponential (eq 26), the
Boens et al.
802 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
F = ko, + k,,
Markov parameters are given by
where r j denotes the ith power of yj. As the Markov parameters are derived fromf(Xcm,Xex,t), they are dependenton the excitation and emission wavelengths. The Markov parameters mican also be written as
mi = K C A ’ ~i, = 0 , 1
~
6
~
)
(40)
mlk = K(416l(kol + k2, + ~ Q I [ Q ]+~ E162ki2 ) E261k21@2(k02 k12 k~z[Qlk)) (41) with 6 2 = 1 - 8, and E2 = 1 - E l . Note that mo is independent of the quencher concentration. b. Determination of the System Parameters from the Identifiability Equations. We now investigate which informationabout the rate constants can be obtained from eqs 35 and 36. From eq 35 at two quencher concentrations [Qlt and [Qll one obtains the following equations:
A = &,I
G = ko,
(39)
The specificexpressions for mo and mlk at quencher concentration [Qlk are then
mo = ~ ( 2 ~+6 ~,
on condition that kQl # kQ2. Combination of B and F (eqs 43 and 48) yields
+k ~ 2
+ k12
=B-F
(49)
and combining E, F,and G (eqs 46, 48, and 49) leads to
FG- E = k2,k12 (50) The three eqs 48-50 are not sufficient to determine the four unknown rate constants kol, kzl, k02, and klz of the system. Therefore, the knowledge of one rate constant is n d e d to determine a complete set of values of rate constants. However, there are two possible ways in eq 45 to combine the quenching rate constants with the other rate constants leading to two sets of values for the rate constants. If kQ1 = kQ2(=kQ) eq 45 can be rewritten as
= (&Ol
+
+ k02 + kI2)kQ
= BkQ
(51)
Furthermore we have
Equation 36 at three quencher concentrations [Qlk, [Q],, and [Qlmyields the following equations:
= ‘QIkQ2
A = 2kQ
(52)
C = k,2
(53)
kQcan be calculated from eq 51 or 52 or 53. The other four rate constants kol, kZ1,k02, and k12must be determined from B and E (eqs 43 and 46), which is impossible even when one of these rate constants is specified. This situation is comparable to the intramolecular two-state excited-state system without added quencher.6 Equation 43 (B) corresponds to -uI whereas cq 46 (E) corresponds to u2 of the bicompartmental system without added quencher. Therefore, the system is only identifiable upon the conditions which are valid for the case without quencher.6 If kQ2equals zero eqs 42, 44, and 45 simplify to eqs 54, 55, and 56, respectively
(45)
A = kQl
(54)
c=o
(55)
(kO2
+ kl2)kQl
(56)
G (eq 49) and F (eq 48) are then given by
= Q2k - D[Q]k
- c[Qlk2
(46)
The quenching rate constants can be determined from A (eq 42) and C (eq 44). It should be pointed out that one of the quencher concentrationsmay be equal to zero. This means that the decay times of the unquenched bicompartmental system together with those at two quencher concentrationssuffice to determine values of the quenching rate constants. Once values for kQland kQ2are specifically assigned, one can proceed as follows. Combining B and D (eqs 43 and 45) produces D = (k01 + k21)(kQ2- kQl) + BkQl from which one obtains that
G = DIA
(57)
F= B-G
(58)
Similar equations can be derived for the case where kQl = 0. The cases in which one quenching rate constant vanishes do not require a separate identifiability study. Assuming that values are assigned to all rate constants (vide infra), 6, and El can be determined as follows. For the decay trace at quencher concentration [Q]r collected at Aem due to excitation at XQ, elimination of the scaling factor K from q s 40 and 41 leads to eq 59
[6I(pk - Qk) + Qkl = 61(R- sk) + sk
(47)
with
(59)
The Journal of Physical Chemistry, Vol. 97, No.4, 1993 803
Intramolecular Two-State Excited-State Processes
zero. Indeed, if m~or mlk at quencher concentration [Q]k is q u a l to zero, eq 59 and the equations derived from it cannot be built. If mo = 0, the precxponentialsal and 4 2 (eq 26) have the same absolute value but opposite signs, Le., a1 = - a 2 or al/a2 = -1. Then the following equality is satisfied:
6,- 1 E , =26,- I
(76)
Since both 6, and E l are normalized (eqs 27 and 28), we have that 0 I61I1 and 0 5 El 5 1 and consequentlyeq 76 is fulfilled only in two physically feasible cases, namely when (1) 8, = 0 and E, = 1 or (2)6, = 1 and El = 0. These conditions are fulfilled when only one species in the ground state is excited and the fluorescence of the other species is recorded exclusively. For example, for a monomer/excimer bicompartmental system where only the monomer is excited and where only the excimer fluorescence is measured, we have that 01, = -a2, indicating that mo = 0. Fluorescence decay measurements where a, = 4 2 do not provide extra kinetic information to extract the values of the system parameters. mlk equals zero at quencher concentration [Q]k if alyl = - a 2 7 2 or T , / T ~ = -aI/a2 # 1, and then the following equality is true:
E, = [ai(k21 + k02 + k12 + kp[Qlt) - (k02 + k12 + k~2[Qlk)l/ [ ~ I & O I + 2k21 + k02 + 2k12+ ( ~ Q+ I k~,)[Qlk)- (k02 + From two decay traces corresponding to the quencher concentrations [Q]k and [Q], recorded at ACm due to excitation at Acx we can construct two q s 59 with identical 61 and El. Elimination of 6, from these equations leads to a quadratic equation in El
EI2T(k,1)
+ E,U(k, 1) + V(k, 1) = 0
(68)
with
V(k, 1)
R(S1- Sk)
(71)
Elimination of El from the same two eqs 59 leads to a quadratic equation in 6,
6,2X(k,1)
+ 6,Y(k, 1) + Z(k, 1) = 0
(72)
with
Z(k, 1) = Q$I - Q S k
(75)
Note that T(k, I), U(k,l), V(k, l), X(k, l), Y(k,I), and Z(k, 1) depend on the excitation and emission wavelengths also. Furthermore, they are proportional to the difference ([Qb- [Qll). Consequently, the eqs 68 f o d as a function of quencher concentration are linearly dependent. The same is true for the eqs 72. The use of eq 59 and the equations derived from it is bound by the following limiting condition: the Markov parameters of the considered fluorescence decay must both be different from
2k12 k~,[Qlk)] (77) The cases where mlk equals zero (TI/TZ = 4 1 / a 2# 1) depend on the set of rate constants and on the quencher concentration [Q]k (eq 77), and they occur very infrequently in real physical systems compared to the cases where mo equals zero (al/az = -1). If 6, at Acx is known and different from zero and one, eq 72 at two quenchcr concentrations, [Qlk and [Q],, can be used to determine the rate constant values. Similarly,if El at XCmis known and different from zero and one, eq 68 at two quencher concentrations,[Q]kand [Qll,provides the necessaryinformation to obtain the rate coefficients. To mathematically determine sets of values of the system parameters, the knowledge of one of the system parameters which may not be &QI or &Q* is necessary. In the next sections we shall discuss the different alternatives. 1. One Rate Constant ki,Is Known. If one rate constant is known (e& kol through the use of a suitable model compound) the two sets of remaining three rate constants (k21, k02, and k12) can be determined using q s 48-50 provided that I # & ~ 2For . each of these sets of rate constants the values for I and El can be determined as follows. From two decay traces corresponding to the quencher concentrations [Q]k and [Q]Irecorded at ACmdue to excitation at Acx a quadratic equation in 61 (eq 72) can be derived. The solution of eq 72 yields two values for 61. As mentioned before, including a third decay trace corresponding to quencher concentration [Qlm does not provide independent information. El can be calculated from eq 59 for each acceptable 6,value. Alternatively, a similar approach can be followed for E l starting from eq 68. The previous approach builds on measurementsdone at a single excitation and emission wavelength and leads to four sets of solutions for the system parameters. To reduce the number of solutions one should proceed as follows. To obtain the single set of values for the rate constants, decay traces of a suitable model compound measured at a minimum of two quencher concentrationsmust be included in the analysis. As before, one of the quencher concentrations may equal zero. In this way the correct (ko,,k~,)association is established.
P
804
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
The unique set of (81,E~) values can be recovered by including one or more decay curves at additional Xcx (with different 81) or Xcm (with different E l ) with mo # 0. Indeed, enclosing an additional excitation or emission wavelength leads to an extra independent eq 68 or eq 72, respectively. If the singly known rate constant is k21 and equals zero, the value of k02 or klz must be available. If the singly known rate constant is k12and equals zero, either kol or kll must be known. These preconditions are evident by inspection of eqs 48-50. When either kzl or k12equals zero, eq 50 reduces to FG-E=O (78) and theknowledgeofoneadditional rateconstant isa prerequisite to estimate the remaining system parameters. If none of the rate constants of the intramolecular bicompartmental system is known, one 8, or one 21 value must be specified to determine the possible sets of system parameters. This is inve_tigated in the next t,wo sections. 2. One Value Is Known. If bl at XeX is known and different from zero and one, eq 72 at two quencher concentrations, [Q]k and [Qll, can be rearranged to
Note that H , Z, J , K,and L depend not only on [Qlk, [Q], but also on XeXand Xem. Furthermore, they are proportional to (kQ2 - k ~ ~ ) ( [ Q-l k[Qll), leading to the following two consequences. Switching the (kQl,kQ2)association leads to the interchange of kQ!and kQz in eqs 80-84 and thus produces an identical eq 79. Using a third quencher concentration produces a linearly dependent eq 79. Equations 48-50 and 79 comprise a system of four equations in four unknowns. Equations 48,49, and 79 allow us to express kol, k21, and k02 as a function of k12. Substitution of k21 in eq 50 gives the following quadratic equation in k12
kl,t(K-J)+kI2(FH+GJ-L)-(FG-E)(H-I) = O (85)
Alternatively kol, k02, and k12may be expressed as a function of k2l. Substitution of kl2 in eq 50 yields a quadratic equation in k21. Once a nonnegative value for k12is obtained from eq 85, kol. kzl and koz may be calculated from eqs 48-50. El at X C is~ calculated from eq 59. Since eq 85 is quadratic, two sets of rate constants are mathematically possible. As twoequationsof type 85 are possible by interchanging the values for ~ Q and I k ~ in 2 F (eq 48), a total
Boens et al. of four sets of system parameters are mathematically possible. The inclusion in the analysis of decay traces collected for two quencher concentrations at an additional emission wavelength with a different El creates an extra eq 79. The system of four equations linear in the rate constants (eqs 48, 49, and twice eq 79) reduces the number of possible sets of system parameters to two. If the fluorescencedecaysurface includes at least nine decay traces measured at four emission wavelengths with different El (corresponding to at least three quencher concentrations at the first emission wavelength and at least twoquencher concentrations at the other three emission wavelengths), the unique set of system parameters will beobtained. Indeed, for each emission wavelength with different El at two quencher concentrations, an independent eq 79 can beconstructed. The system of four linearly independent eqs 79 has a unique solution. The unique value of F (eq 48) leads to the proper correspondence between k~~and koi. It is evident that, when a decay trace with mo = 0 is included in the global bicompartmental analysis, the information that 61 is one (when E l = 0) or zero (when El = 1) may not be used. 3. Oae E1 Value Is Known. If E l at XCm is known and different from zero and one, eq 68 at two quencher concentrations, [QIk and [Q]I,can be rearranged to eq 79 with
Here also H, Z,J , K,and L depend not only on [Qlk, [Q]Ibut also on Acx and Xcm. Furthermore, they are proportional to (kQ2 ~ Q I )[Qlk ( - [Qll), having the following two consequences. Switching the (kQl,kQ2)association leads to the interchange of k ~ and l k ~ in 2 eqs 86-90 and thus produces an identical q 79. Using a third quencher concentration produces a linearly dependent eq 79. The rate constants kol, k21,~ Q Ik02, , k12,k ~ 2and , 81at Xcxcan be calculated by a procedure which is similar to that explained in the previous section where one 81value is known. In analogy with the previous case (where 81 is known), the inclusion in the analysis of decay curves measured for at least two quencher concentrations at a supplementary excitation wavelength with a different 6, reduces the number of possible sets of system parameters to two. At least four excitation wavelengths with different 8 , are necessary to obtain the unique set of system parameters. It is evident that when a decay trace with mo = 0 is included in the global bicompartmental analysis, the information that El is one (when 81= 0) or zero (when 81 = 1) may not be used. c. Summary of tbe Identifubnty Study. The results of the identifiability study can be summarized as follows: for an intramolecular (i.e., concentration independent) system with added quencher Q as depictedin Scheme I the following conditions have to be fulfilled in order to obtain a finite number of sets of system parameters. First, the fluorescence decay surface must include at least one set of decay traces measured at a minimum of three different quencher concentrations for the same excitation/emission wave-
Intramolecular Two-State Excited-State Processes
The Journal of Physical Chemistry, Vol. 97, No. 4, 1993 805
TABLE I: (A) Values of the Rate Coastrats Utilized in the Simulation of the Bicomprtnmtal System Depicted in Scheme I. (B) Alternative Set of Rate Constants for Known’ kol = 5 x 107 s-I k21= 60 x 107 S-I kQI 200 X lo7 M-l
A
kO2= io x 107 S-I k12= 8 X lo7s-I kQ2 = SO X 10’ M-I
S-I
B
kol = 5 x 107 s-I k21= 13 x 107 s-I kQI = SO x 107 M-1 s-I
S-I
kO2= 28 x 107 S-I kI2= 37 x 107 s-I kQ2 = 200 X lo7 M-I
S-I
See text for details.
TABLE 1l: Preexponential Factors a1 and az,and Decay Tima -yl-l and q 2 - I (in ns) as a Function of E1 Calculated for the Different Quencher Concentrations (in M) with the Rate Constants of Table IA’ 0
0 0.2 0.8 0 0.2 0.8 0 0.2 0.8
0.1
0.2
a &I
-0.61 -0.38 0.32 -0.53
0.9 1 0.76 0.29 0.83 0.68 0.24 0.75 0.62 0.20
1.36
1.09
-0.30 0.38 -0.45 -0.29 0.41
0.90
10.67
6.23
4.50
= 0.7 in all cases.
TABLE IIk- Estimated Parameter Values of the Rate Coastants, b~and 4, from Global Bicompartmental Analysis when bl (5 x 107 9-1) IS kol(107 s-1) kll (107s-1)
[Qli
Q-IM-I)
kO2(107s-9 kI2(io7s-I) k ~ (lo7 2 8-1 M-I)
5(4 59.8 f 0.8 202 f 8
10.01 f 0.07 7.8 f 0.9 49 f 2
5 (4
28 f I
kQi
81
(M)
PI’
0 0.1 0.2
0.2 0.18 f 0.03 0.68 f 0.04
0 0.1 0.2
0.2 0.94 ( c )
0.03 (c)
0 0.I 0.2
0.2
0.21 f 0.02
0 0.1 0.2
0.2
El
(lo7
A
B
measured at four emission wavelengths with different E l (corresponding to at least three quencher concentrations at the first emission wavelength and at least two quencher concentrationsat the other three emission wavelengths), to uniquely determine the set of system parameters. If the known system parameter is a SIvalue different from 0 and 1. the fluorescence decay surface must includeat least nine decay traces measured at four excitation wavelengths with different 6, (corresponding to at least three quencher concentrations at the first excitation wavelength and at least two quencher concentrationsat the other three excitation wavelengths), to uniquely determinethe set of system parameters. In no case the addition of a decay trace where mo = 0 provides indispensableinformation. Negativevalues for the rate constants, and values for 6, and El larger than 1 or smaller than 0 may reduce the number of acceptable sets of system parameters. 4. 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 analysisprogram2 based on Marquardt’sloalgorithm. Any or all decay parameters can be kept fixed during the fitting or may be freely adjustable to seek optimum values. Consider the intramolecular two-state excited-state process depicted by Scheme I. The global fitting parameters are ko1,k z l , ~ Q Iko2, , klz, kQ2, &(Aex), and El(ACm). The only local fitting parameters are the scaling factors K’. Specifying the quencher concentrations and assigning initial guesses to the rate constants kol,k21,~ Q Ik02, , k12, and kQ2allows one to construct the compartmental matrix A for each decay trace. The eigenvalues y and the associated eigenvectors of this matrix are determined using routines from EISPACK, Matrix Eigensystem Routines.II The eigenvectors are then scaled to the initial conditions 6. The preexponential factors are computed using eqs 25-28. The fluorescence response of the sample, ds(t), is calculated according eq 31. Using this approach, experiments done at different excitation/emission wavelengths, at multiple timing calibrations, and at different quencher concentrationsare linked by all rate constants defining the system. The fitting parameters were determined by minimizing the global reduced chi-square xg2
C 0.87 f 0.02
12f2 49 f I
36fI
5 (4 I2fI
28f 1 36 f 1 202 f 8
202 f 8
D 0.08 f 0.01
0.95 f 0.01
49fI
The simulation values of the rate constants are compiled in Table IA and the target value of &I = 0.7.
length setting. One of the used quencher concentrations may be q u a l to zero. Second, the two rate constants of quenching must bedifferent. Third, at least onesystemparameter must be known. Under these conditions four sets of system parameters are mathematically possible. If the known system parameter is a rate constant (kol, k21,k02, or k12. but not ~ Q orI kQ2)r decay traces of a suitable model compound measured at a minimum of two quencher concentrations must be included in the analysis in order to obtain the unique set of values for the rate constants. The unique set of ( & , E l ) values can be recovered by including decay curves at a minimum of two quencher concentrations and at an additional excitationwavelength with a different or at another emission wavelen th with a different E l . If the known system parameter is a I value different from zero and unity, the fluorescence decay surface must include at least nine decay traces
d
where the index I sums over q experiments and the index i sums over the appropriatechannel limits for each individualexperiment. y ” ~and pii denote respectively the observed (experimentally measured or synthetic) and calculated values corresponding to the ith channel of the lth experiment,and wri is the corresponding statistical weight. v representsthe number of degrees of freedom for the entire multidimensionalfluorescence decay surface. The statistical criteria to judge the quality of the fit comprised both graphical and numerical tests. The graphical methods included plots of surfaces (“carpets”)of the autocorrelationfunctionvalues vs experiment number, and of the weighted residuals vs channel number vs experiment number. The numerical statistical tests incorporated the calculation of the global reduced chi-square statistic xe2 and its corresponding ZX;,
Using Zx8zthe goodness of fit of analyses with different v can be readily compared. b. Synthetic Data Generation. Synthetic sample decays were generated by convolution off(r) with a nonsmoothed measured instrument response function u ( t ) . Various experimental u ( t ) were used in the simulations. The preexponential factors ai and corresponding eigenvalues yi of the biexponential decays were
806 The Journal of Physical Chemistry, Vol. 9 7 , No. 4, I 9 9 3
Boens et al.
TABLE IV- Estimated Parameter Values of the Rate Constants, 4 and 4, from Global Bicompartmental Analysis When the Decay Trace of an Unquenched Model Compound Is Added to the Three Decays of Table 111'
[Qli
(MI
81
i.1
?I'
kol (IO7 s-I) k2l ( I O 7 s-l) kQi (lo7
M-I)
s-I
k02 (IO7 s-I) klz (lo7s-I) kQz (lo7 s-I M-I 1
A 0.2 0.18 f 0.03 0.68f 0.04 4.999 f 0.005 10.01f 0.07 59.8 f 0.8 7.8 f 0.9 202 i 8 49f 1
0
0.1 0.2 model
B* 0 0.2 0.94 ( c ) 0.1 0.2 model
A
0.03 (c)
5 (c)
10 ( c ) 8 (4 50 (4
60 (4 200 ( c )
C 0.2 0 . 8 7 f 0.02 0.21 f 0.02 4.999f 0.005 2 8 f 1 12f2 36i 1 49f 1 202 f 8
0 0.1 0.2 model
D 0 0.2 0.08 f 0.01 0.95 f 0.01 4.999f 0.005 2 8 f 1 0.1 12f 1 36 f 1 0.2 49f 1 202 f 8 model W
+-J
v'
, Q
B Figure 1. Zx8: surfacesobtained from the global bicompartmental analyses of three decays at three different quencher concentrations where the rate constants, bl and E l , were kept fixed during the analysis. In part A the rate constants are taken from Table IA and the two minima correspond to the analyses of Table IIIA,B. Part B was generated using the rate constants of Table IB and the two minima correspond to the analyses of Table IIIC,D. The values of 51and i.1 were varied between zero and one in increments of 0.05.
computed from the rate constants k,, 81,and ?I by a dedicated computer program. The preexponential factors were adjusted to obtain the desired number of counts. All computer simulated decays had l/zK data points with between 5 X lo3 and 2 X lo4 counts in the peak channel. The time increment per channel (20, 30, 40, 50, 100, or 150 ps) was chosen to ensure that the final decay intensity was about 5% or less of the initial intensity. Full details of the decay data simulations are given elsewhere.12 The synthetic data generations and all individual and global analyses were done on an IBM RISC System/6000 computer. c. Parameter Estimation from Computer-Generated Decay Data. To test the validity of the conclusions of the identifiability study, computer-generated decay data were used. The values of the six rate constants of the bicompartmental system depicted in Scheme I utilized in the simulations are compiled in Table IA. The preexponential factors aiand decay times calculated with these values at the various quencher concentrations [Q] and ZIare shown in Table 11. When necessary, additional simulations were done to illustrate most of the special cases which were discussed in the identifiability study. For example, a second set of simulations were generated with 6, = 0.8 and k12= 0. The values for the remaining rate constants were taken from Table IA. The true (target) values for the fitting parameters are indicated in Tables 111-VI1 by a superscript t. A value followed by (c) means that this parameter is kept constant at the given value during the analysis. In Tables 111-V a capital letter followed by an asterisk means that this set of system parameters could not be recovered unless all linked parameters were kept constant at
The simulation values of the rate constants are compiled in Table IA and the target value of 81 = 0.7.
-
TABLE V: Estimated Parameter Values of the Rate from Global Bicompartmental Analysis Constants, 4 and i?~, Where the Three Decays of Table I11 Were Added to Three Decay Traces of a Model Compound Quenched at the Same Quencher Concentrations'
[Qli (M)
i.1'
i.1
6,
kol (IO'S-') kil (lo's-') kQi (10' s-I M-I
)
k02 ( ~ O ' S - ' ) k12(10'~-') k ~ (210' s-I M-I1
A
0 0.2 0.19f0.03 0.70f0.02 4.999f0.005 10.01 f0.05 0.1 59.7f 0.8 8.0f 0.6 0.2 200.0f0.1 49f 1 quenched model
B* 0.2 0.94 ( c )
0 0.1
0.2
quenched
0.03 (c)
5 (c)
10 (4
60 (4 200 (c)
8 (c) 50 (c)
model
The simulation values of the rate constants are compiled in Table IA and the target value of 61 = 0.7. a
their target values. The recovered values different from the simulation parameters will be indicated in the tables in italics. All quoted errors are 1standard deviation. The initial parameter guesses for the global compartmental analysis were substantially different from the true (simulation) parameter values. The different situations, as discussed in section 3, will now be investigated using simulated decay data sets. 1. One Rate Constant Is Known. As shown in the identifiability l kp, is known, study, if one rate constant different from k ~ and four sets of system parameters are mathematically possible. The altemativeset of the rateconstants arecalculated by interchanging the values of kQland kQ2 in D (eq 45 or 47). The alternative set of rate constants for the specific case when kol is known is given in Table IB. For each set of the rate constants, the two sets of ( & , E l ) values are obtained from eqs 68 and 59 or from eqs 72 and 59 for given mo, mlk, mil, Ulk, u11,U2k, and u21. This accounts for the four possible sets of system parameters which can beobtained
Intramolecular Two-State Excited-State Processes
The Journal of Physical Chemistry, Vol. 97, No. 4, I993 807
TABLE VI:- Estimated Parameter Values of the Rate Constants, b, and E l , from Global Bicompartmental Analysis Where the Three Decays of Table 111 Were Added to Three Decays Recorded at a Second Emission Wavelength (with El = 0.8) and Three Decay Traces of a Model Compound Quenched at the Same Quencher Concentrations*
[Qll ( M )
i.1'
61
i.1
kol (10' s-I) k2l (lo7 s-I) kQi (10' s-I M-I)
k0l
TABLE VI& Estimated Parameter Values of the Rate Constants, b, and E l , from Global Bicompartmental Analysis of Six Decay Traces a,t Three Quencher Concentrations and Tyo El Values When b, Is Kept Constant at Its True Value (b, = 0.7)
( lo7 s-I)
k l l (IO7 SKI)
k ~ (2I O 7 s-I M-l
)
0 0.2 0.19f0.03 0.70f0.01 4.999f0.005 10.01 f O . 0 4 0.1 0.8 0.78f 0.01 59.6 f 0.5 7.7 f 0.5 0.2 200.0 f 0.1 50 f 1 quenched model
The simulation values of the rate constants are compiled in Table I A and the target value of 61 = 0.7.
[QI! (MI 0 0.1 0.2
0 0.1 0.2
TABLE VII: Estimated Parameter Values of the Rate Constants, b, and 81, from Global Bicompartmental Analysis of Simulated Decay Data Using the Rate Constants of Table IA but with k12 = 0'
0 0.1
0.8 0.7f 0.3
0.8f 0.2
30 f 40 30f40 0.2 207 f 4 0 0.8 0.805 f 0.005 0.75f 0.03 5 (c) 0.1 59 f 4 0.2 207 f 4 0 0.8 0.805 f 0.004 0.76 f 0.03 5 (c) 0.1 59.2f 0.5 0.2 207 f 4 The target value of
10.01 f 0.03 O(c) 50.5 f 0.4 10.00 f 0.04 (2 f 4)10-4 50.6 f 0.4 10.01 f 0.03 0 (c) 50.5 f 0.4
81 = 0.8
by changing the initial guesses in the least-squares search of the fluorescence decay surface. The results with a known value of kol are shown in Table 111. We chose to keep kol constant because this is the rate constant which is the most likely to be known from a suitable model compound. Similar results were obtained by fixing other rate constants (results not shown). Table IIIA-D shows the four sets of system parameters recovered from three decay traces at the indicated quencher concentrations. Figure 1A shows the Zx2 values for global compartmental analyses of three decays atthree different quencher concentrations as a function of (bl,Z!)
i.1
Of6 63 f 6 197 f 7
k02 (lo7s-1) kiz (lo7 Swi) k ~ (2IO7 S-I M-I) 10.6 f 0.8 7.9 f 0.7 49.2f 0.6
TABLE I X - Estimated Parameter Values of the Rate Constants, b, and El, from Global Bicompartmental Analysis of Six_Decay Traces at Three Quencher Concentrations and Two 4 Values When 4 Is Kept Constant at Its True Value ( E l = 0.2)
[Qli (MI
Figure 2. Zxg2 surface obtained from the global bicompartmental analyses where the three decays of Table 111 were added to three decays recorded at a second emission wavelength (with i.1 = 0.8) and three decay traces of a model compound quenched at the same quencher concentrations. The rate constants taken from Table IA, bl and i.1 were kept fixed during the analysis. The single minimum corresponds to the analyses of Table VI. The values of 61and i.1 were varied between 0 and 1 in increments of 0.05.
21'
0.2 0.21 f 0.03 0.8 0.8 f 0.1
kol(107 s-ljkzl(107 s-1) kQ1 ( l o 7S-I M-l)
kol (107 S-I) kZI(107 s-1) 81' 81 kQi (lo7S-' M-') 0.4 0.40f 0.06 4f8 0.7 0.70 f 0.06 60 f 9 199 f 6
ko2 ( l o 7 s-I) k12 (lo7s-I) kQ2 ( I O 7 s-I M-I) 10f 1 8f2 49f 1
corresponding to the rate constants of Table IA whereas Figure 1B was generated using the rate constants of Table IB. The four sets of system parameters arevery clearly indicated by the minima of the Zx,2surfaces in Figure 1A,B. In Table IV we considered the case where a monoexponential decay trace with kol = 5 X lo7 s-' ( 7 = 20 ns) is added to the three biexponential decays of Table I11 with [Q] = 0, 0.1, and 0.2 M. This case corresponds to the availability of an appropriate model compound with kolidentical to that of the bicompartmental system. Then it is not necessary to keep kol constant at its known value. It suffices to link kol over all measured decay traces. The effect of this addition is similar to fixing kol at its simulation value (see Table 111). The monoexponential decay of the model compound determines unequivocally the value of kol. As shown in Table 111, this addition still allows for four sets of values for the system parameters. However, when the decays of a quenched model compound are incorporated (Table V), the analysis will yield the set of rate constants of Table IA. Indeed, not only the values of kol and k~~ are determined but also the correct assignment is made. The minima in the Zxg2surface for the second set of rate constants (Table IB) disappear, while the Zxgz surface for the first set of rate constants (Table IA) still shows the same two minima as in Figure 1A. Finally, adding three decay traces at a second emission wavelength to the six decays of Table V leads to the unique set of system parameters (Table VI). In Figure 2a single minimum is observed in the Zxg2surface corresponding to the simulation parameters. Table VI1 shows the results of the analysis of the simulations with k12 = 0. It has been shown in the identifiability study that the information k12= 0 is not sufficient to determine all system parameters. Table VI1 shows indeed that fixing k12at zero gives imprecise and inaccurate values for kol and k l l . The knowledge of one additional rate constant is required to obtain accurate and precise paymeter estimates. 2. One 4 Value Is Known. The identifiability study predicts that, if one 6 , is known and different from zero and one, four complete sets of system parameters are mathematically possible using threedecay traces measured at a single emission wavelength and corresponding to three different quencher concentrations. The number of possible sets of system parameters can be reduced to two by adding three decay traces recorded at a second emission wavelength with different SI. The unique solution (Table VIII) is obtained because the alternative set of system parameters is physically unacceptable. The accuracy and precision of the recovered rate coefficients are substantially decreased compared to the case when a rate constant is known.
808 The Journal of Physical Chemistry, Vol. 97, No. 4, 1993
3. One Value Is K”.If one El is known and different from 0 and 1, the identifiability study predicts four sets of system parametersusing three decay traces measured at a single emission wavelength and corresponding to three different quencher concentrations. The number of possible sets of system parameters can be reduced to two by adding three decay t r a m recorded at a second excitation wavelength with different 8,. The unique solution (Table IX)is obtained because the alternative set of system parameters is physically unacceptable. Compared to the case when a rate constant is known, the accuracy and precision of the recovered rate coefficients are significantly decreased.
Ackuowledgment. N.B. is a Bevoegdverklaard Navorser of the Belgian Fonds voor Geneeskundig Wetenschappelijk Onderzoek. B.H. is a predoctoral fellow of the Instituut tot Aanmoediging van het WetenschappelijkOnderzoek in de Nijverheid en de Landbouw (Belgium). R.A. acknowledges the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek for a predoctoral fellowship. The continuing support of the Ministry of Scientific Programming through UIAP-11- 16 is gratefully acknowledged.
References and Notes (1) (a) Knutson, J. R.; Beechem, J. M.; Brand, L. Chem. Phys. Lett. 1983,102,501-507.(b) Mfroth, J.-E. Eur. Biophys. J. 1985,13,45-58.(c)
Boens et al. Beechem, J. M.; Ameloot, M.; Brand, L. Anal. Instrum. 1985,!4,37942. (d) Beechem. J. M.: Brand. L. Photochem. Phorobiol. 1986.44.323-329. (el Jansens, L. D.; Boens. N.; Ameloot. M.; De Schryver, F. C. J: Phys. Ch&. 1990, 94,35643576. (2) Boens,N.;Janssens, L. D.; DcSchryver, F. C. Biophys. Chem. 1989, 33.77-90. (3) (a) Becchcm, J. M.; Ameloot, M.; Brand, L. Chem. Phys. Lett. 1985, 120,466472. (b) Amcloot, M.; Beechem, J. M.; Brand, L. Chem. Phys. Lett. 1986, 129,211-219. (4) (a) Ameloot, M.; Bans, N.; Andriessen, R.; Van den Bcrgh, V.; De Schryver, F. C. J . Phys. Chem. 1991, 95,2041-2047. (b) Andriessen, R.; Boens, N.; Ameloot, M.;De Schryver, F. C . J. Phys. Chem. 1991,95,20472058. (5) Andriessen. R.;Ameloot, M.; Boens, N.; DcSchryvcr. F. C . J . Phys. Chem. 1992, 96,314-326. (6) Boens, N.; Andriessen, R.; Ameloot, M.; Van Dommelcn, L.; De Schryver. F. C. J. Phys. Chem. 1992, 96,63314342, (7) Rabcnstein, A. L. Elementary D@erential Equations with Unear Algebra, 3rd ed.;Harcourt Brace Jovanovich: San Diego, CA, 1982. (8) (a) O’Connor, D. V . ; Phillips, D. Time-correlated single photon counting, Academic Press: London, 1984. (b) Bans, N.In Luminescence Techniques in Chemical and Biochemical Analysis; Baeyens, W.R.G., De Keukeleire, D., Korkidis, K.. Eds.;Marcel Dekker: New York. 1991;p 2145. (9) Anderson, D. H. Compartmental Modeling and Tracer Kinetics; Lecture Notes in Biomathematics; Springer-Verlag: Berlin, 1983. (IO) Marquardt, D.W.J. Soc. I d . Appl. Math. 1963, 11, 431-441. ( 1 1) Smith, B. T.; Boyle, J. M.;Garbow, B. S.;Ikeke, Y.; Klema, V. C.; Molcr, C. B. In Lecture Notes in Computer Science; Goos, G., Hartmanis, Eds.; SDrinacr Vcrlaa: Hcidelbern. Germanv. 1974 Vol. 6. (1 2j V& den &&I, M.; Boen6N.; Dams; D.; De Schryver, F. C. Chem. Phys. 1986, 101, 311-335.
