J. Phys. Chem. 1994,98, 8585-8590
8585
Determination of the Ground-State Dissociation Constant by Fluorimetric Titration Andrzej Kowalczyk,+NoGl Boens,' Viviane Van den Bergh, and Frans C. De Schryver Department of Chemistry, Katholieke Universiteit Leuven, 8-3001 Heverlee, Belgium Received: October 25, 1993; In Final Form: June 8, 1994"
The fluorimetric determination of the ground-state dissociation constant Kd of intermolecular bicompartmental systems in the absence and presence of an excited-state reaction is discussed. The presence of a n excited-state reaction can result in very complicated fluorimetric titration curves, precluding the evaluation of Kd. The interference of the excited-state reaction can be circumvented if the fluorimetric titration is performed a t the isoemissive point. Under this condition, the correct value of Kd will be obtained from the unique inflection point of the titration curve. It is further shown that previous recommendations to excite either a t the isosbestic point or a t a wavelength where only one species absorbs are inappropriate.
SCHEME 1
Introduction The recently developed probes for the fluorimetric nondestructive determination of intracellular calcium, sodium, potassium, and magnesium ions and of intracellular pH1 have both facilitated and stimulated extensive new research in avariety of areas. These indicators change their absorption and/or emission properties upon binding to the ion of interest. Spectrophotometry and fluorimetry represent powerful tools for the evaluation of groundstate dissociation constants and the determination of the ion concentration. The advantages of fluorescence over absorption measurements comprise higher sensitivity (because fluorescence is detected us a dark background) and selectivity (one may avoid the signal from other absorbing molecules). Less probe is required in fluorimetry than in spectrophotometry to attain a similar sensitivity. Fluorimetry generally results in less disturbance of the system and allows less soluble probes to be used. Fluorescence can also be used with samples of high turbidity. Furthermore, the geometrical requirements for fluorescence detection are less stringent than for absorption. However, serious difficulties can arise from the dependence of fluorescence on the ground-state equilibrium and on all rates of the excited-state processes. The problem of interference of excited-state reactions in the determination of the ground-state equilibrium constant has not been given much attention by the pioneers of the analytical application of fluorescence.*4 The significance of that complication was first realized by Schulman and co-workers,*~6but their analysis appears to be inaccurate. In this paper, weshallderive theequationsrelating themeasured fluorescencesignal to the concentration of the titrant in the absence and presence of the excited-state reaction. It will be shown that the interference of the excited-state reaction can be avoided when the fluorescence is observed at the isoemissive point. The previous recommendationssq6to excite at an isosbestic point or in the range where only one of the forms of the indicator absorbs are shown to be improper. Since compartmental analysis has been very successful in analyzing the time-resolved fluorescence originating from intermolecular two-state excited-state processes,7 we shall adopt the terminology and methodology of compartmental analysis to describe the corresponding steady-state fluorescence. For a photophysical definition of the term compartment we refer to ref 8.
Ground-State Equilibria Determined by Fluorescence spectroscopy Without Excited-State Reaction. Consider a causal, linear, time-invariant, concentration-dependent (i.e. intermolecular) system consisting of two distinct types of ground-state species (i.e. a bicompartmental ground-state system) and two corresponding distinct types of excited-state species (Le. a bicompartmental excited-state system). The kinetic model for such a system is presented in Scheme 1. In the ground state, species 1 can undergo a reversible reaction with X to form species 2. Species 1represents the free form of a fluorescent indicator, while species 2 the corresponding bound form. If Scheme 1 depicts the ionization of a weak acid (species 2), species 1 is the conjugated base of the weak acid, and X is the solvated hydrogen ion. It is assumed that only species 1 and 2, which are in chemical equilibrium with each other, absorb light at the excitation wavelength hex. The molar extinction coefficients for species 1 and 2 are denoted by el(Xex) and t 2 ( X C X ) , respectively. Scheme 1 assumes a 1:l stoichiometry between species 1 and X. 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 kol ( = k ~ 1 +klcl+ kIsc1) and k02 ( = k ~ 2 + k 1 ~ +2 k&. No excited-state reaction is assumed in this kinetic scheme. For the compartmental system depicted in Scheme 1, the measured steady-state fluorescence signal F(Xex,hem, [XI) due to excitation at hex and observed at hem is given by7 F(XeX, Xem, [XI) = -e(Xem) c(lem)A-' b(XCX,[XI)
(1)
where t(Xem) is an instrumental factor. c is the 1 X 2 vector of spectral emission weighting factors cj(XCm),
* To whom correspondence should be addressed. f
On leave from the Institute of Physics, Nicholas Copernicus University,
87-100 Torun, Poland. 0
Abstract published in Advance ACS Abstracts, July 15, 1994.
kFj is the fluorescence rate constant of species i*; Ahcm is the emission wavelength interval where the fluorescence is monitored;
0022-3654/94/2098-8585$04.50/0 0 1994 American Chemical Society
8586 The Journal of Physical Chemistry, Vol. 98, No. 34, 1994
Kowalczyk et al.
SCHEME 2 (3)
where the integration extends over the whole steady-state fluorescence spectrum Fi of species i*. A is the compartmental matrix for the system7**
(4) and A-1 its inverse. b(Aex, [XI) is the 2 X 1 vector of the zerotime concentrations [i*](O). If Beer's law is obeyed and if the absorbance of the solution is low (> Kd, only species 2 will be present, and since it is not excited, no fluorescence will be observed for any observation wavelength. It is noteworthy that the above remains true when [XI= Kd*. Upon lowering [XIin the concentration region of Kd, the ground-state concentration of species 1will increase while that of species 2 will decrease. Since only species 1 is excited and since [XI Kd*, only species 2 and 2* will be present. If the fluorescence of only species 1* is monitored, the fluorescence signal will drop to zero. When Xem is set tocollect the fluorescence of species 2* exclusively, the fluorescence signal will reach its maximal value. From the above discussion and Figure lb,c, it is clear that when pKd* and pK# > pKd, the titration curves exhibit a unique inflection point at [XI = Kd only in the case when €1 = 0 M-' cm-1. For the case when €2 = 0 M-l cm-l, multiple inflection points are observed. One of the inflection points corresponds to pK#. It must be emphasized that the titration curves collected at the isoemissive point (bold curves in Figure 1b,c) yield a unique inflection point at [X] = Kd. Figure 2b,c shows that when pKd* and pK# < pKd, the titration curves exhibit a unique inflection point at [XI = Kd only in the case when €2 = 0 M-I cm-I. For the case when el = 0 M-1 cm-1, multiple inflection points are observed. One of the inflection points corresponds to pKd, the second pK#. When a third inflection point is found, it is located between the two previous ones. Again, the titration curves
q
I
I
a
\
I
o
Y
i
PK,
PKX
PK,'
Figure 3. Calculated fluorimetric titration curves for the hypothetical system using the rate constant values of Figure 1. Excitation at hem where t l = 10 000 M-' cm-* and €2 = 30 000 M-I cm-1). The numbers = 1. The curve corresponding on the curves are the CI values used. to the isoemissive point is drawn in bold.
collected at the isoemissive point (bold curves in Figure 2b,c) yield a unique inflection point pKd. It is clear that the recommendation536to excite at a wavelength where one of the species does not absorb presupposes the apriori knowledge of the relative values of pK# and pKd. Therefore, this procedure should always be avoided. The only approach to determine the ground-state dissociation constant Kd from fluorimetric titration is to measure the titration curves at the isoemissive point. This procedure can be performed at any excitation wavelength which does not correspond to the isosbestic point. This is illustrated in Figures 1 and 2. If the fluorimetric titration is not done at the isoemissive point, the following situations may lead to confusion and/or misinterpretation: (i) one inflection point occurs at pK# (Figure la,c); (ii) two inflection points occur which cannot unequivocally be assigned top& and pK# (Figures IC and 2a,b); (iii) two inflection points occur which can be due to two titratable groups. The confusion and/or misinterpretation can be avoided by performing the fluorimetric titration at the isoemissive point. Indeed, for situations i and ii the unique inflection point will occur at pKd, while the ambiguity of the origin of the two inflection points in caseiii will beremoved.I1 Toshow the widevariety of fluorimetric titration curves, wegenerated curves with the rate constant values of Figure 1 and chose an arbitrary excitation wavelength where €1 = 10 000 M-1 cm-1 and €2 = 30 000 M-1 cm-1 (Figure 3). It is obvious that only the curve measured at the isoemissive point shows a unique inflection point which corresponds to the correct valueofKd (0.99-1.00). Theother titrationcurvesexhibit peculiar shapes with multiple inflection points. We haveclearly demonstrated the pivotal role of theisoemissive point in deriving the correct Kd from fluorimetric titrations when an excited-state reaction is present. It is possible to determine the isoemissive point from steady-state fluorescence spectra when the fluorescence spectra and quantum yields of the two species can be measured separately. Alternatively, upon excitation at an isosbestic point (el = Q),the observed crossing point of the emission spectra of samples with the same CTand different [XI will occur at a true isoemissive point. Indeed, it can be shown that eq 6 or 9 with ai given by eq 14 is independent of [XIonly when el = €2 and cllkol = c2/k02. In contrast to the case with no excited-state reaction, excitation at any wavelength different from an isosbestic point will not produce a common crossing of the fluorescence spectra. The lack of a common crossing in such a case (see for example ref 12) can be due to excitation at a wavelength different from an isosbestic point or to other factors. Excitation at an isosbestic point will remove this ambiguity. If the two above methods fail to determine the isoemissive point, global compartmental analysis of time-resolved experiments'
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The Journal of Physical Chemistry, Vol. 98, No. 34, 1994
provides the method for its determination. If, however, no isoemissive point exists, the correct value Of Kd must be evaluated from spectrophotometric titrations. According to Beer’s law, absorbance is an additive property of all absorbing species present in the solution. The total absorbance A at Acx for a given [XIshould thus be equal to the sum of the absorbances of the individual species present.
or through the use of the relationships of eq 8 one has
The concentration dependence of A (eq 2 3 ) has the same functional form as that of F in the absence of an excited-state reaction (eq 9 with ai defined by eq 7 ) . Therefore, mutatis mutandis the discussion which follows eq 9 applies here as well. It should be emphasized that the inflection point of the spectrophotometric titration curve will yield the correct value of Kd. The lower sensitivity and selectivity of absorption us fluorescence measurements often make spectrophotometric titrations inferior. Moreover, in order to obtain the same sensitivity as offered by fluorescence, the amount of probe must be increased, possibly resulting in higher disturbance of the system. The higher concentrations necessary in spectrophotometric titrations exclude the use of indicators with low solubility.
Conclusions The presence of an excited-state reaction can obstruct the determination of the ground-state dissociation constant Kd by fluorimetric titration. The interference of the excited-state reaction can be avoided if the fluorescence is observed at the wavelength corresponding to the isoemissive point. Under this
Kowalczyk et al. condition, the correct value of Kd will be obtained from the unique inflection point of the titration curve. It is further shown that the previous recommendations to excite either a t the isosbestic point or at a wavelength where only one species absorbs are inappropriate.
Acknowledgment. A.K. had an Individual E.C. Fellowship under the Community mobility action for Cooperation inscience and Technology with Central and Eastern European Countries which enabled him to stay at the K.U. Leuvan. H e also thanks the K.U. Leuven for the hospitality during his stay there. N.B. is an Onderzoeksleider of the Belgian Fonds voor Geneeskundig Wetenschappelijk Onderzoek. V.V.d.B. is a predoctoral fellow of the Znstituut ter Aanmoediging van het Wetenschappelijk Onderzoek in de Nijverheid en de Landbouw, Belgium. The continuing support of the Ministry of Scientific Programming through IUAP-11- 16 is gratefully acknowledged. References and Notes (1) Haugland, R. P. In Handbook of Fluorescent Probes and Research Chemicals, 5th ed.; Larison, K. D., Ed.; Molecular Probes: Eugene, 1992. (2) Weller, A. Prog. React. Kinet. 1961, 1, 189-214. (3) Ireland, J. F.; Wyatt, P. A. H. Adu. Phys. Org. Chem. 1976, 12, 13 1-221. (4) Schulman, S. G.In Modern Fluorescence Spectroscopy; Wehry, E. L., Ed.; Plenum: New York, 1976; Vol. 2, pp 239-275. ( 5 ) Rosenberg, L. S.;Simons, J.; Schulman, S. G.Talanra 1979, 26, 867-871. (6) Rosenberg, L. S.; Lam, G.;Groh, C.; Schulman, S.G. Anal. Chim. Acta 1979, 106, 81-87. (7) Ameloot, M.; Boens, N.; Andriessen, R.; Van den Bergh, V.; De Schryver, F. C. J. Phys. Chem. 1991, 95, 2041-2047. (8) Boens, N.; Andriessen, R.; Ameloot, M.; Van Dommelen, L.; De Schryver, F. C. J. Phys. Chem. 1992, 96,63316342. (9) Van den Bergh, V.; Boens, N.; De Schryver, F. C.; Ameloot, M.; Gallay, J.; Kowalczyk, A. Chem. Phys. 1992, 166, 249-258. (10) Salinas, F.; Mufioz de la Pefia, A.; Mufioz de la Pefia, F. Anal. Lett. 1989, 22, 2371-2380. (11) Valant, P. A.; Adjei, P. N.; Haynes, D. H. J . Membr. Biol. 1992,130, 63-82. (12) van Stam, J.; Lafroth, J.-E. J. Chem. Educ. 1986, 63, 181-184.
