Introduction

Acridines and acridones are widely used compounds in analytical and medicinal chemistry [1]. These heterocycles also show a great importance in the field of molecular recognition as they are often used as units of several host molecules, due to the effective coordinating ability of their nucleophilic nitrogen atom toward various cations and their advantageous properties like structural rigidity, strong fluorescence, photostability, etc. [1]. The first acridino- and acridono-crown ethers were synthesized in the late 1990 s [2]. Since then, several sensor molecules containing an acridone or an acridine unit have been synthesized for various applications [3,4,5,6,7,8,9,10,11,12,13,14]. As we are interested in the synthesis and application of new acridino- and acridono-host molecules, a study on the application influencing properties and their molecule structural background plays an important role in the effective future development.

Acridine is a weakly basic (pKa (conjugate acid) = 5.4 in water) [15] heterocycle. Oxidation of acridine gives stable acridin-9(10 H)-one, shortly called acridone, which contains a carbonyl group at position 9 of the heterocycle. Previous studies showed that in the case of acridono-macrocycles tautomerism takes place upon the complexation with cationic species [16, 17]. The host molecule suitable for coordination is in its hydroxyacridine tautomeric form in the complex [16, 17]. Reduction of acridine gives acridane, which is an unstable compound, thus its derivatives are generally not suitable for practical applications. Regarding structural features, hosts containing an acridine unit are considered to be the most advantageous ones for recognition of cations, because of the nucleophilicity of the nitrogen and the ability of the electron rich tricyclic system for π-interactions. On the other hand, acridines containing a hydrogen atom at position 9 have a high tendency to oxidation, thus their conversion to the stable acridones takes place during the applications. Oxidation of the acridine unit can be avoided by the substitution of the hydrogen atom at position 9 resulting in a stable derivative.

Regarding various applications, the focus of the development of crown ether type sensor molecules can be traditionally divided in two large categories: optochemical- and electrochemical ones. Although in the case of electrochemical sensors the signaling is not directly determined by the molecular form of the heteroaromatic unit, in the case of optical signaling the outstanding quantum yield of the acridone form results in an increased signal strength in most cases. In practice, the ionophores are usually immobilized physically or covalently into synthetic membranes, which provides the regenerability and the stable operation of the obtained analytical devices. The high lipophilicity of acridino-crown ethers (logP ~ 8 predicted by the logD Calculator Plugin of ChemAxon for the parent acridino-18-crown-6 ether) gives an opportunity for physical immobilization in apolar membranes. Moreover, the macrocycle can be tailored to further enhance the lipophilic character or to arm with reactive groups for covalent embedding. Since these modifications of the oligoether unit usually have a weak effect on molecular recognition of achiral guests, parent crown ethers are suitable model compounds for general studies on the effects governing applications.

Molecular recognition-based sensor applications are influenced by several factors, like the counterion of the ionic guest, the solvent, the polarity of the medium, the temperature, etc. Thus, it is difficult to reveal general structure-activity relationships, even in a relatively small group of sensor molecules as crown ethers containing an acridone or an acridine unit. However, studies can be simplified by neglecting certain conditions for example polarity-dependent factors, since in most of the practical applications the ionophores are embedded in apolar membranes and the analyte is in an aqueous solution. Similarly, the measurements are usually carried out at room temperature. The main factor that is always present due to the generally examined aqueous samples is the pH of the medium, in which the measurements are carried out. Protonation of the nitrogen of the sensor molecules and their prototropic conversion, if it takes place, have strong effects both on complexation and signaling. As it was previously mentioned, the electrochemical signaling is not directly effected by the molecular form of the heterocyclic unit in the hosts, the spectrophotometric methods provide more information about the signaling properties of these sensor molecules. Therefore, the molecular recognition abilities of crown ethers containing an acridone or an acridine unit were optochemically investigated from the aspect of the relationship between complex formation and protonation, since they can be considered as the most important characteristics effecting the practical application of the fluorescent hosts as sensor molecules.

The key structural transformations and equilibrium processes during the applications of sensor molecules containing an acridone or an acridine unit are summarized in Scheme 1.

Scheme 1
scheme 1

The typical structural transformations during the practical applications of crown ether type host molecules containing an acridone or an acridine unit (The studied processes and model compounds were indicated in red and the curved arrows represent the equilibrium shifting effects)

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The present studies were carried out using the examples of a Pb2+-selective 9-phenylacridino-18-crown-6 model compound (as an example for compound 3 see Scheme 1) and its Pb2+-selective acridono-analogue 4 (detailed information regarding the molecular recognition abilities of the applied sensor molecules is reported previously [18, 19]). This comparison study can provide generalizable relationships for the effective future development of ionophore-based cation sensors containing an acridone or an acridine unit.

Results and discussion

Spectrophotometric studies on acridono-crown ether type macrocycles

Initially, acid titration of acridono-crown ether 4 as a Pb2+-selective sensor molecule was performed by UV/Vis- and fluorescence spectroscopies for studying acridono-macrocycles in acidic conditions (Fig. 1). Crown compound 4 was dissolved in acetonitrile, as it is practically insoluble in water. Solutions of concentrated nitric acid in acetonitrile were added in 0–40 molar equivalents to crown ether 4.

Fig. 1
figure 1

Series of spectra for UV/Vis- and fluorescence titrations of acridono-crown ether 4 with nitric acid solutions (0, 0.01, 0.05, 0.1, 0.2, 0.4, 0.6, 0.8, 1, 2, 3, 4, 10, 20, 40 eq) (a: Changes in absorbance in the presence of acid, b: Titration spectra at two different characteristic absorption wavelengths of the macrocycle, c: Changes of fluorescence in the presence of acid, d: Titration spectrum at the emission peak wavelength of the macrocycle)

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It can be seen that the absorbance decreased upon addition of the acid at the wavelength of the absorption maximum (251 nm), while at 265 nm (wavelength at the maximum of the other characteristic absorption band) an increase in absorbance was observed (see part a and b in Fig. 1). The nitrogen atom of the acridone unit does not show a basic character. (The pKa values of some macrocyclic analogues of crown ether 4 were previously determined [20].) Thus, the observed spectral changes in acidic conditions were probably caused by solvatochromic effects. This assumption is consistent with the fact that the photophysical properties of acridine derivatives are generally extremely sensitive even to small polarity changes of the medium [21]. In the case of fluorescence titration, a slight decrease in the intensity of emission was observed, which can also be attributed to solvatochromic effects (see part c and d in Fig. 1). However, partial protonation of host 4 cannot be totally excluded based on the results.

The addition of 200 molar equivalents Pb2+-perchlorate (the end point of Pb2+-titration based on previous studies on the same crown ether [18]) to the solution of crown ether 4 resulted in a significant spectral change upon complex formation (Fig. 2).

Fig. 2
figure 2

Absorption spectra of free acridono-host molecule 4 and its Pb2+-complex (a, ccrown ether=0.5 µmol/L) and the tautomeric transformation of the heteroaromatic unit of macrocycle 4 upon complexation (b)

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The spectral change indicated that the electronic structure of the host molecule was also altered and the acridone unit transformed to its 9-hydroxyacridine tautomeric form due to complexation (see part b in Fig. 2).

Therefore, when the Pb2+-complex of crown ether 4 was titrated with an acid, different spectral changes were observed compared to those of the free host molecule with a heteroaromatic unit in its acridone form (in the case when the tendency to form a complex is stronger than that is for protonation). The 9-hydroxyacridine tautomeric form of complexed host molecule 4 contains an acridine-like nitrogen atom, which shows a weakly basic character [15]. In order to compare the spectra of the acid titration of the free host and the complex, the Pb2+-complex of crown ether 4 was also titrated with nitric acid in the same way as previously described (Fig. 3).

Fig. 3
figure 3

Series of spectra for UV/Vis-titration of the Pb2+-complex of acridono-crown ether 4 with nitric acid (a: Changes in absorbance in the presence of 0–0.8 eq acid, b: Changes in absorbance in the presence of 0.8–40 eq acid, C: Changes in absorbance by adding nitric acid to the complex at its absorption maximum)

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In the case when acid was added to the complex an increase in absorbance was observed up to the addition of 0.8 molar equivalents of it (see part a of Fig. 3). In contrast, additions of 0.8 to 40 equivalents of acid caused quenching in absorbance (see part b of Fig. 3). It suggests that the tautomeric equilibrium of the heteroaromatic unit shifts toward the acridone form due to the protonation of the 9-hydroxyacridine tautomer. At the end point of the titration (after addition of 40 eq acid) the spectrum returned to that of a similar one of free acridono-macrocycle 4 (see part c of Fig. 3).

The same titration procedure was also carried out using fluorescence spectroscopy (Fig. 4).

Fig. 4
figure 4

Series of spectra for fluorescence titration of the Pb2+-complex of acridono-crown ether 4 with nitric acid (a: Changes in fluorescence in the presence of 0–0.8 eq acid, b: Changes in fluorescence in the presence of 0.8–40 eq acid, C: Changes in fluorescence intensity by adding nitric acid to the complex at its emission maximum)

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The results of the two measurements were consistent. Trends in the series of spectra changed direction after the addition of 0.8 molar equivalents of acid.

Spectrophotometric studies on acridino-crown ether type macrocycles

Studies were carried out in the same manner using the example of Pb2+-selective 9-phenylacridino-crown ether derivative 3 as a model compound for acridino-type crown ethers. The spectra obtained from the titration of the free macrocycle with nitric acid are shown in Fig. 5.

Fig. 5
figure 5

Series of spectra for UV/Vis- and fluorescence titrations of acridino-crown ether 3 with nitric acid (a: Changes in absorbance in the presence of acid, b: Changes in absorbances at the two different characteristic absorption bands of the macrocycle, c: Changes in fluorescence intensity in the presence of acid, d: Changes in fluorescence intensity by adding nitric acid to macrocycle 3 at the emission maximum)

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Protonation resulted in an isosbestic point in the absorption spectrum (see parts a and b of Fig. 5), while it causes an almost total quenching in fluorescence (see parts c and d in Fig. 5). In addition to it, a bathochromic shift was also observed (446 nm → 486 nm). The spectral change as a function of the amount of the added acid was significantly greater in the case of acridino-macrocycle 3 than that for acridono-crown ether 4. It suggests that the crown ether stabilized by substitution at position 9 in its acridine form is more sensitive to acidic medium due to its increased basicity compared to the acridone form.

Subsequently, 200 molar equivalents of Pb2+ were added to the solution of acridino-macrocycle 3. The resulted spectrum is shown in Fig. 6.

Fig. 6
figure 6

Absorption spectra of free acridino-host molecule 3 and its Pb2+-complex (a, ccrown ether=0.5 µmol/L) and structures of the free crown ether and its Pb2+-complex (b)

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The observed significant spectral change indicated complex formation.

After addition of 200 molar equivalents of Pb2+ the resulting complex was titrated with nitric acid (Fig. 7).

Fig. 7
figure 7

Series of spectra for UV/Vis- and fluorescence titrations of the Pb2+-complex of acridino-crown ether 3 with nitric acid (a: Changes in absorbance in the presence of acid, b: Changes in absorbance at the absorption maximum of the complex, c: Changes in fluorescence intensity in the presence of acid, d: Changes in fluorescence intensity by adding nitric acid to the complex at the emission maximum)

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In contrast to acridono-macrocycle 4, the absorbance decreased gradually, thus presumably the protonation of a single molecular form took place (see part a and b in Fig. 7). In the case of fluorescence measurements the results (see part c and d in Fig. 7) were close to those obtained during the protonation of free acridino-host molecule 3 (see part c and d in Fig. 5). It also suggests that the protonation of the acridino-crown ethers is less effected by the complexation than that of the acridono-analogues.

Changes in the pK a of protonated macrocycles upon complex formation

As it was mentioned earlier, the nitrogen atom of acridono-crown ethers is not basic. In the case of these macrocycles, complexation induces tautomerism of the heterocyclic unit, which has been previously confirmed by IR- and 1 H-NMR spectroscopies and single-crystal X-ray diffraction [16, 17]. These studies - in accordance with the present work - support that the 9-hydroxyacridine tautomeric form is preferred during the complex formation. Theoretically, the pKa of the protonated acridono-crown ether complex could be determined by the protonation of the basic nitrogen of its 9-hydroxyacridine tautomeric form (5). However, based on this study it is assumed that the heteroaromatic unit is converted back to its tautomeric acridone form upon protonation, which inhibits complex formation. Hence, the pKa for this complex equilibrium system cannot be determined spectroscopically due to the copresence of the acridone tautomeric form.

Determination of the pKa values for protonated acridino-crown ether 3 and its Pb2+-complex (regarding the deprotonation process of the protonated nitrogen of the acridine moiety of the macrocycle) was performed based on the calculation described in Experimental section. A global nonlinear regression method was carried out and the pKa values were determined based on a mathematical function fitted according to the least square’s method over the entire wavelength range of the titration spectra. The fitted curves for crown ether 3 and its complex are shown in Fig. 8.

Fig. 8
figure 8

Fitted nonlinear regression curves for pKa determinations of the conjugate acid of acridino-crown ether 3 (a) and its Pb2+-complex (b) based on spectroscopic titration data

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The pKa values of protonated acridino-crown ether 3 and its Pb2+-complex were found as 8.86 ± 0.19 and 7.96 ± 0.22, respectively. It can be seen that the pKa of the protonated macrocycle decreased by approximately one unit as a result of complex formation. In the case when the lone pair of the nitrogen atom of the host molecule is involved in the coordination with the guest molecule, the basicity of the heteroatom decreases. The change in the pKa value also shows that protonation and complex formation are in a competitive relationship with each other.

Based on our previous experience, the pH-independent working range of the membrane-embedded acridono-macrocycles applied as both optical and electrochemical Pb2+-sensors ranged from the neutral pH to 4.0–4.3 [22, 23]. In more acidic conditions, protonation of the nitrogen of the crown ethers caused deviance in the response of the sensors. Detection was carried out in aqueous medium in all cases, in which the conjugate acid of the parent acridine has a pKa of 5.4 [15]. In the presence of Pb2+, the acridono-crown ether is expected to behave (according to the pKa) similarly to the protonated acridine when the pH of the medium changes. From this assumption it can be stated that the pKa values of acridono-macrocycles in their hydroxyacridine forms do not change in the presence of Pb2+. Thus, for acridono-crown ethers, complexation would not inhibit protonation and vice versa. However, it is only a presumption. When acridono-crown ethers were used as optochemical Pb2+-sensors, we have previously hypothesized that there should be a competition between complex formation and protonation of the macrocycles. This competition can manifest via the shifting of the tautomerization equilibrium and the decreasing electron donating ability of nitrogen upon the coordination of metal ions. According to the Henderson-Hasselbalch equation, a significant proportion (above 10 %) of a compound with a pKa of 5.4 should already be in its protonated state at pH 6.3, which should cause a significant signal change of the sensors. Thus, the pH-range for application could only be extended to about 6.3 from 7.0. Since this limit was 4.0-4.3, we can conclude that the pKa of the protonated macrocycle in the presence of Pb2+ decreased by approximately 2 units (although, due to the physical immobilization of the macrocycles, tautomeric conversion may be inhibited and the equilibrium may also be effected by the altered polarity of the medium). As mentioned previously, it is not possible to determine the pKa of the protonated hydroxyacridine tautomer of acridono-crown ethers (5) based on spectrophotometry. The magnitude of the decrease in pKa values upon complexation for acridono-macrocycles can be estimated as 2 units based on the above considerations and previous studies on these types of sensor molecules [22, 23].

Comparison of complex stability constants in neutral and acidic media

Acridono-crown ether 4 was titrated with Pb2+, and then the Pb2+-titration was also carried out in the presence of 40 molar equivalents of acid (corresponding to the end point of the acid titration of the macrocycle). Results are shown in Fig. 9.

Fig. 9
figure 9

Series of spectra for fluorescence titrations of acridono-macrocycle 4 with Pb2+ in neutral (a) and acidic conditions (b)

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In both cases, nearly total fluorescence quenching was observed at the end point of the titration. In the case of neutral medium, the end point of the titration was reached by adding 10,000 molar equivalents of Pb2+, while in the case of acidic conditions more than one order of magnitude molar equivalents of Pb2+ was needed for it. Consistently, the rate of spectral change per unit of added Pb2+ also proved to be greater in neutral medium. All these prove that the host molecule forms a more stable complex in the absence of acid. In order to quantify the stability of the complexes global nonlinear regression method was similarly used to determine stability constants based on spectroscopic titration data. The fitted global nonlinear curves for Pb2+-titration data of acridono-crown ether 4 are shown in Fig. 10.

Fig. 10
figure 10

Fitted nonlinear regression curves for logK determinations of the Pb2+-complex of acridono-crown ether 4 both in neutral (a) and acidic (b) conditions based on spectroscopic titration data

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The relationships used for calculations can be found in Experimental section. The logK for the Pb2+-complex of acridono-crown ether 4 was 3.71 ± 0.17 in neutral medium (this logK was previously determined by Benesi-Hildebrand method based on UV/Vis titration and was found 3.64 [18]) and 2.22 ± 0.20 in acidic conditions (the logK determined in acidic conditions is an apparent constant). Results show that in acidic conditions, the logarithm of the complex stability constant is less by 1.49. This can be attributed to shifting the tautomeric equilibrium toward the acridone form in acidic conditions, which tautomer of the heterocyclic unit is unfavorable for complex formation.

Stability constants were also determined for crown ether 3 containing an acridine unit in the same conditions. Pb2+-titration in neutral medium for crown ether 3 (see part a of Fig. 11) was previously carried out and the complex stability constant was determined as logK = 5.07 [19]. The results of Pb2+-titration of this crown ether in acidic conditions (after the addition of 40 molar equivalents of acid corresponding to the end point of the acid titration) are shown in part B of Fig. 11.

Fig. 11
figure 11

Series of spectra for fluorescence Pb2+-titrations in neutral and acidic conditions for studying the complex stability toward Pb2+ in the case of the unprotonated (a) and protonated (b) acridino-macrocycle 3

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It can be seen in part b of Fig. 11 that there was no spectral change upon addition of Pb2+, thus complex formation did not take place. This is because the protonation of the acridine unit inhibits the intermolecular interactions with Pb2+.

In summary, the pH-dependence of complexation is much less in the case of crown ethers containing the heterocyclic unit in acridone form (logK (Pb2+, neutral medium) = 3.71 ↔ logK (Pb2+, acidic medium) = 2.22) than for macrocycles containing stabilized acridine forms (logK (Pb2+, neutral host) = 5.07 ↔ logK (Pb2+, protonated host) ≈ 0). This can be attributed to the possibility of tautomerism. The complexation of Pb2+ is more strongly inhibited in the case of the stabilized acridine form than that of crown ethers containing an acridone unit. The inhibitory effect of protonation could be reduced by tautomerization.

Studies showed that only the acridine form of the heterocyclic unit took part in the complexation with the metal ion. If no prototropic equilibrium takes place due to the substituent at position 9 of the heterocyclic unit, a higher complex stability was obtained toward Pb2+ (logK (acridono-crown ether 4) = 3.71 ↔ logK (acridino-crown ether 3) = 5.07). However, in the cases of both crown ethers (host molecules 4 and 3) a high competition took place between protonation and complex formation. In the case of acridono-macrocycle 4, the acidic medium partially suppressed the conversion to the preferred 9-hydroxyacridine tautomer (5), while the stabilized acridino-crown ether (3) was unable to coordinate Pb2+ due to the protonation of its nitrogen atom. It also indicates the key role of the nitrogen atom among the other donor atoms of the macrocycle in molecular recognition.

Conclusions

Comparison of the reported types of sensor molecules was carried out using hosts 3 and 4 as model compounds, which are suitable candidates for general studies on factors influencing practical applications. In the case of acridino-crown ethers the acid inhibits complex formation more strongly (ΔlogK (logK change upon acidifying) for the complexes of acridino-crown ether ≈ 5 ↔ ΔlogK for the complexes of acridono-crown ether = 1.49), while in the case of acridono-macrocycles the complexation of the guest cation inhibits protonation more strongly (ΔpKa (pKa change upon addition of Pb2+) for acridono-crown ether ≈ 2 ↔ ΔpKa for acridino-crown ether = 0.9) and vice versa. Therefore, in the case of the acridono-macrocycles, complexation is less sensitive to acid and the pH-sensitivity of the crown ethers decreases even more after complexation. There is a weak competition in this case. The pKa of the protonated acridino-crown ether (3) changes only half as much upon complex formation than it does of the hydroxyacridine tautomer (5) of acridono-crown ethers (4). There is a strong competition. Complexation is more strongly inhibited in this case and the presence of the preferred guest cation can only slightly compensate the effect of the acidic medium, even that the acridino-crown ether (3) forms a complex with more than 1 order of magnitude higher stability toward Pb2+ compared to the acridono-analogue (4).

Therefore, in terms of pH-sensitivity, crown ethers containing an acridone unit are more advantageous. From an application point of view, the pH of the medium is one of the most limiting conditions effecting the practical use of the macrocycles. This inhibitory effect can easily be limited in the case of the hydroxyacridine form of the studied heterocyclic units. Thus, crown ethers containing an acridone unit are recommended for the development of cation-complexation-based applications. Naturally, the macrocycles containing a heteroaromatic unit in a stabilized acridine form could also be advantageous, especially for selector applications in the case of organic guests. In this case a fixed and stabilized acridine form is required, which can be obtained by substitution at position 9 of the heterocycle. However, the change in the pH of the medium can be a strongly limiting factor for complex formation.

Experimental

Materials and solvents were purchased from Sigma-Aldrich Corporation (USA, owned by Merck) and their purity was analytical grade. Macrocyclic model compounds were prepared as reported [19, 24]. UV/Vis spectra were recorded on a UNICAM UV4-100 spectrophotometer controlled by VIZION 3.4 software (ATI UNICAM, UK). Fluorescence emission spectra were recorded on a Perkin-Elmer LS 50B luminescent spectrometer (PerkinElmer Inc., USA) and were corrected by FL Winlab 3.0 spectrometer software (PerkinElmer Inc., USA). Quartz cuvettes with path length of 1 cm were used in all cases. The concentrations of the solutions of macrocycles were 1.0 µmol/L in acetonitrile for the acid titrations and 0.5 µmol/L in acetonitrile for the titrations with Pb2+. For spectrophotometric titrations, the hydrogen perchlorate salts of Pb2+ in acetonitrile or concentrated nitric acid in acetonitrile were added to the solutions containing the crown ether using a Hamilton-syringe. The results were corrected with the background signal and the dilution effect of the added solutions. Every titrant solution was added until constant optical signal was observed which state was defined as the end point of the titration experiment. In the case of fluorescence measurements an excitation wavelength of 251 nm was used for the acridono- and 265 nm for the acridino-model compound. The complex stability constants and the pKa were also determined in acetonitrile at 25 ± 1 °C.

Determination of pKa values in non-aqueous medium was carried out based on the following equation [25]:

$$F= \frac{{F}_{max}{\left[{H}^{+}\right]}^{n}+{F}_{min}{K}_{acid}}{{K}_{acid}+{\left[{H}^{+}\right]}^{n}}$$
(1)

where F is the measured fluorescence intensity, Fmax is the fluorescence intensity at the starting point of acid titration, [H+] refers to the proton concentration, n shows the number of associated proton / molecule, Fmin is the fluorescence intensity at the end point of acid titration, Kacid is the acid dissociation constant of the investigated compound. The pKa values were determined by global nonlinear regression analysis using OriginPro 8.6 (OriginLab Corp., USA) software. During the fitting method, the n and the Kacid were defined as parameters in the equation. The value of n proved to be close to 1, thus it was set as a constant. Hence, based on the known values of variable [H+] and the wavelength-dependent variables F, Fmax, Fmin, parameter Kacid can be determined. During the calculation it was considered that the proton dissociation of nitric acid is strongly reduced in acetonitrile compared to the estimated total dissociation of protons in water. The pKa for nitric acid in acetonitrile is 10.6 [26]. The [H+] values in Eq. 1 were corrected with the degree of dissociation corresponding to the concentration of nitric acid using the Ostwald’s dilution law.

The stability constants of the complexes were also determined by global nonlinear regression analysis using the same software. For determination of the complex stability constant based on the observed fluorescence quenching by complexation, the following equation was used [27]:

$$\frac{{F_{0} - F}}{{F_{0} - F_{c} }} = \frac{{\left[ H \right] + \left[ L \right] + \frac{1}{{K_{a} }} - \sqrt {\left( {\left[ H \right] + \left[ L \right] + \frac{1}{{K_{a} }}} \right)^{2} - 4 \cdot \left[ H \right] \cdot \left[ L \right]} }}{{2 \cdot \left[ H \right]}}$$
(2)

where F is the measured fluorescence intensity, F0 is the starting fluorescence of the host molecule, Fc is the fluorescence of the fully complexed host, Kα the association constant of the complex, [H] the concentration of host, [L] the concentration of the added ligand at a given point of titration. For easier handling, the equation has been aligned to F:

$$F = F_{0} - \left( {F_{0} - F_{c} } \right) \cdot \frac{{\left[ H \right] + \left[ L \right] + \frac{1}{{K_{a} }} - \sqrt {\left( {\left[ H \right] + \left[ L \right] + \frac{1}{{K_{a} }}} \right)^{2} - 4 \cdot \left[ H \right] \cdot \left[ L \right]} }}{{2 \cdot \left[ H \right]}}$$
(3)

F 0 is a wavelength-dependent variable and [L] was set as a variable, too. During fitting, Fc and Kα were defined as parameters, while the values of [H] were given as constants.