Applying non-parametric statistical methods to the classical

Anal Bioanal Chem (2003) 375 : 414–423 DOI 10.1007/s00216-002-1693-0

O R I G I N A L PA P E R

E. Almansa López · J. M. Bosque-Sendra · L. Cuadros Rodríguez · A. M. García Campaña · J. J. Aaron

Applying non-parametric statistical methods to the classical measurements of inclusion complex binding constants Received: 10 July 2002 / Revised: 4 November 2002 / Accepted: 6 November 2002 / Published online: 30 January 2003 © Springer-Verlag 2003

Abstract A study on using non-parametric statistical methods was carried out to calculate the binding constant of an inclusion complex and to estimate its associated uncertainty. First, a correct evaluation of the stoichiometry was carried out in order to ensure an accurate determination of the binding constant. For this purpose, the modified Benesi-Hildelbrand method had been previously applied. Then, four statistical methods (three non-parametric methods: two bootstrap approaches, the jackknife method and a parametric one: Fieller’s theorem) were employed in order to compute the binding constant. The results obtained from applying these methods and the combination of the methods: jackknife after bootstrap and bootstrap after jackknife were compared. The best results in terms of accuracy were obtained from the application of a bootstrap method: the resampling residuals approach. These procedures were applied to the inclusion complex 2-hydroxilpropyl-β-cyclodextrin-2,4-dichloro-phenoxyacetic, which shows photochemically-induced fluorescence. Electronic Supplementary Material Supplementary material is available for this article if you access the article at http://dx.doi.org/10.1007/s00216-002-1693-0. A link in the frame on the left on that page takes you directly to the supplementary material. Keywords Inclusion complex · Benesi-Hildebrand method · Bootstrap · Jackknife · Fieller’s theorem

E. Almansa López · J. M. Bosque-Sendra (✉) · L. Cuadros Rodríguez · A. M. García Campaña School of Qualimetrics, Department of Analytical Chemistry, University of Granada, 18071 Granada, Spain e-mail: [email protected] J. J. Aaron Institut de Topologie et de Systèmes de l’Université Denis DIDEROT, Paris 7, Laboratoire Associé au CNRS, Paris, France

Introduction Cyclodextrins [1] (CDs) comprise a family of three wellknown industrially produced major, and several rare, minor cyclic oligosaccharides. The three major CDs (α-, βand γ-cyclodextrin) are crystalline, homogeneous, nonhygroscopic substance, which are torus-like, macro-rings built up from glucopyranose units. In an aqueous solution, the slightly apolar cyclodextrin cavity is occupied by water molecules which are energetically unfavorable (polarpolar interaction), and therefore can be readily substituted by appropriate “guest” molecules (substrate) which are less polar than water. The dissolved cyclodextrin is the “host” molecule and the most frequent host : guest ratio is 1:1. In the case of organic molecular structures showing fluorescence, their interaction with CDs usually produces an increase in the fluorescence signal [1, 2]through a partial encapsulation or total inclusion. The intensification of luminescence processes of certain molecules included in the interior of the cavity of the CD is due to the high protection from quenching and the other processes occurring in the bulk solvent. Nevertheless, the limiting solubility of CDs in water is one parameter that prevents fluorescence enhancement. In spite of its low solubility in water, β-CD is the most widely used, since the diameter of its cavity is usually compatible with most of the compounds [2, 3]. Consequently, the structure of β-CD has been modified in order to improve its solubility and prevent its crystallization. For example, 2-hydroxypropyl-β-CD (HP-β-CD) [3, 4] presents a much greater solubility (1620 mg/ml at 25 °C in a 50/50 v/v water/methanol mixture) than β-CD (13 mg/ml in the same mixture). Because of their inherent usefulness in the areas of chemical, pharmaceutical, environmental or food analysis, among others, several studies have been carried out to evaluate their complexing ability [1] of CDs. In order to research and develop new analytical methods for pollutant monitoring, the study of the luminescent properties of the inclusion complexes between CDs and pesticides is cur-

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rently a subject of increasing interest in Analytical Chemistry for their promising perspectives as sensitive detection systems in separation techniques. In the case of pesticide analysis, there are relatively few papers devoted to the study of fluorescent inclusion complexes with CDs [8]. For pesticides showing native fluorescence, such as warfarin [9], bromadiolone [10] and coumatetralyl [11], the corresponding fluorescence intensities are very intense in organic media but in aqueous solutions they suffer quenching effects due to the interaction with the solvent. This problem is solved by means of the inclusion complex formation with the use of CDs (generally β-CD) in aqueous solutions. For pesticides that do not show native fluorescence, photochemical derivatization permits their conversion into fluorescent compounds. This technique named room–temperature photochemically induced fluorescence (RTPIF) is based on the conversion of nonfluorescent analytes into strongly fluorescent photoproducts after direct irradiation with ultraviolet light. From an analytical point of view, it is not necessary to identify the structure of the fluorescent compound formed after irradiation because the PIF signals are reproducible and directly proportional to the concentration of nonfluorescent analyte. As example, chlorophenoxyacid herbicides do not show native fluorescence but as well as other aromatic pesticides, they can be photolyzed into strongly fluorescent photoproducts [8], allowing the establishment of a new method for their quantitative analysis in methanolic medium [12, 13]. Also, the influence of the presence of organized media such as cationic, anionic and non-ionic surfactants has been studied with regard to the PIF properties of these herbicides in static and flowing stream solutions, allowing the use of PIF detection for the sensitive determination of 2,4-Dichlorophenoxyacetic acid (2,4-D) and Mecoprop (MCPP) in a cationic micellar medium, avoiding the need to use organic solvents [14, 15]. We carried out preliminary studies in order to establish the PIF properties of 2,4-D in presence of HP-β-CD. Further analytical applications are being developed. Generally, a CD forms a 1:1 complex with the substrate but it is also possible that a CD molecule can contain one or more guest molecules. The correct evaluation of the stoichiometry is obviously necessary for an accurate determination of the binding constant, which provides a measure of the complex stability, giving a clear understanding of the factors that influence the complexation. The Benesi-Hildebrand [16] (B-H) method may be used to determine the stoichiometry and the binding constant of the CD inclusion complex. This method has been the most popular for more than a half a century, but it shows several disadvantages in its classical application in terms of inaccuracy in the estimation of the binding constant [17, 18]. With the aim of being able to recommend an adequate procedure, in this paper we have evaluated different statistical methods to determine binding constants and their associated uncertainties. For comparison purposes, these methods have been applied to the photochemically induced fluorescent inclusion complex formed by HP-β-CD and 2,4-diclhorophenoxyacetic acid (2,4-D).

The Benesi-Hildebrand Method The equilibrium between a substrate, S, and the CD is: n · S + m · CD ↔ Sn CDm

where m and n are the corresponding stoichiometric coefficients for S and CD, respectively. The inclusion complexes are usually formed by one molecule of CD (m =1, stoichiometry n:1), and therefore the binding constant for SnCD inclusion complex is: Xb [S − CD] K= (1) n= [CD] · [S] (CCD −CS · Xb ) (1 − Xb )n in which [S-CD], [CD] and [S] represent the equilibrium molar concentrations of the complexed S, free CD and free S, respectively. CCD and CS are the analytical concentrations of the CD and S, and Xb = [S-CD]/CS is the molar ratio of the bounded S. Considering that the fluorescence intensity of the free and bounded S are FS and FS-CD, respectively, the fluorescence intensity of the free CD is FCD and F is the observed fluorescence at each HP-β-CD concentration tested for the selected excitation and emission wavelengths. The B-H method is based on the continuous variation of CCD that will always be greater than CS, which remains constant. Then, the following equation can be obtained:   1Fmax 1 1 1 1 1 = CnCD −1 → = + · K 1F 1F 1Fmax 1Fmax · K CnCD (2) where ∆F is the fluorescence intensity due to the inclusion complex, being ∆F = F–FS–FCD = ∆Fmax · (1–Xi), and ∆Fmax is the maximum fluorescence intensity that would be obtained if S was totally complexed by HP-β-CD, being ∆Fmax = FS-CD–FS–FCD. In the original B-H method, 1/∆F is plotted against 1/CCD for 1:1 stoichiometry and against 1/C2CD for 2:1 stoichiometry. According to Eq. 2, a linear relationship is obtained when the value of n is chosen correctly, being 1/∆Fmax the intercept and 1/(K·∆Fmax) the slope. This method suffers two main defects: firstly, it tends to put more emphasis on lower concentration values than on higher concentration values, so the slope is more sensitive to the point corresponding to the lowest concentration; secondly, even if the explanatory variable 1/CnCD could be considered free of error and the whole experimental uncertainty were concentrated into the variable 1/∆F, a great inconvenience is that the estimated value K is calculated as a ratio of two normally distributed but dependent quantities: hence, it has an irregular distribution. Therefore, the linear double reciprocal plots should not be used to quantify the binding constant but it is a good method to estimate the stoichiometry (1:1 or 2:1) of the inclusion complex. The B-H method may be modified [19] multiplying the equation 2 by CnCD . With this modification, a single reciprocal plot is obtained plotting CnCD /1Fvs.CnCD : 1 1 CnCD = + ×CnCD = a + b · CnCD 1F 1Fmax · K 1Fmax

(3)

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where 1/∆Fmax = b is the slope and 1/(K·∆Fmax) = a is the intercept, but the two variables are mutually dependent and the estimate of K = b/a is affected by the same second defect of the original B-H method.

take a value equivalent to nine times its standard deviation. For higher values of g, Eq. 4 is essential.

Bootstrap method

Calculation of the binding constant and estimation of its uncertainty by different methods Fieller’s theorem In the standard least squares algorithm for calculating the fitted regression (Eq. 3), we assume that experimental errors are independent and the variances are not significantly different. In this case, the individual estimated coefficients, a and b, are normally distributed and present a zero bias. However, the ratio between a and b is a different matter because the statistical properties of a ratio of normal variables are difficult to establish. In particular, if the ratio is biased and has skew distribution, accurate confidence limit calculations for the binding constant are not easily obtained. Since the binding constant calculation and its associated uncertainty is a very important topic in the analysis of inclusion complex, it is necessary to apply a statistical method for which the basic requirements for the calculation of these values are minimum. The uncertainty can be calculated from Fieller’s well-known theorem [20]. This theorem can be applied to the quotient of two parameters (a and b), which can be a mean, a difference between means, or regression coefficients from experimental data. In our case, these parameters are regression coefficients (K = b/a) and therefore, when applying Fieller’s theorem, the upper (KU) and lower (KL) limits of the ratio are obtained as:  1 g · cov(b, a) KL , KU = K− 1−g s2a  1/2 $  k 2 cov2 (b, a) 2 2 2 ± s − 2K · cov(b, a) + K sa − g sb − a b s2a

(4) 2,

2

where sa sb and cov(b,a) are the estimated variances of a, b and their covariance respectively, k is the coverage factor [21] with ν degrees of freedom (ν= n-2 being n the number of experimental data) and g: k2 s2 g = 2a (5) a When a shows a high value in relation to its standard deviation, g will have a small value; then g can be neglected and Eq. 4 becomes:
1/2 / KL , KU = K±k s2b −2K · cov(b, a) + K2 s2a a (6) and the term between the parentheses is equivalent to the variance of K,
 2 s2K = s2b −2K · cov(b, a) + K2 s2a a (7) This approximation is adequate if g is less than 0.05, which implies that for a limit of probability at 0.95, a will

In addition, some statistical non-parametric methods based on computer simulations have been developed for robust analysis with minimum assumptions. An example is the method named Bootstrap [22, 23, 24]. It provides measurements of the statistical variation of ratios of regression coefficients and estimates the bias and the standard error. It is based on warranted theoretical assumptions. Taking into account a recent tutorial [24] and our own bibliographical review, the number of applications of this method in chemistry is limited. The bootstrap technique has been applied in the determination of a mixture of Tl(I) and Pb(II) using differential pulse polarography [25], in the study of the spectrum-structure correlation of Cu(II) complexes using spectrophotometry [26], in the estimation of the parameters by non-linear regression from polarographic signals [27], in the determination of the aromaticity of a coal based on neuronal network calibration [28], and in the measurement errors on the uncertainty in bilinear model predictions [29]. The bootstrap method examines the variability of an estimate by using the existing data, together with some assumptions about how they were generated, to produce new, but plausible, “pseudo data sets”. The estimates can be obtained for each “pseudo data set” and then, from these resulting values it is possible to obtain approximations to the statistical characteristics of the original estimates. The basic idea is to simulate repetitions of the experiment and to analyze the data, without warranted assumptions about the frequency distribution of experimental errors, and in this way to observe the statistical variation of the regression coefficients ratios. There are two fundamentally different approaches in applying the bootstrap to linear regression problems [24]. Each approach is described in a linear regression setting, where [(CCD)1, Y1], ..., [(CCD)n, Yn] are observed such that: Yi = a + b · (CCD )i +ei

(8)

where i = 1, ..., n experimental data, Yi = (CCD )i /1Fi , a = (1Fmax · K)−1 , b =1F−1 max and the binding constant is K = b/a. The residuals (ei) produce the simplest estimate of the errors and they are calculated as the observed response minus the predicted response from the fitted regression equation (e = Y − yˆ ), where yˆ is the fitted value calculated according to Eq. 4 substituting the corresponding values of a and b. Resampling cases (RC). New bootstrap datasets are created by sampling independently from the pairs [(CCD)i, Yi], yielding [(CCD )Bi , YBi ] . Simulated values aB, bB and KB are computed from [(CCD )B1 , YB1 ] , ..., [(CCD )Bn , YBn ] in

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the same way that a, b and K were computed from the original data. This simulation process is performed a moderately high number of times (NB). Resampling residuals (RR). The residuals are used to make new bootstrap data sets. By sampling independently from ei makes the new data set yield eBi and by setting: YBi = a + b · (CCD )i +eBi

(9)

Then, simulated values aB, bB and KB of the estimated coefficients may be computed from the bootstrap data [(CCD )i , YBi ] . This simulation process is performed a moderately high number of times (NB). In both cases, the statistics of interest such as the bias and the standard error can be derived from repeated sets of the experimental results:  Q B B 2 Ki −K B s2 (K)B = BIAS(K)B = K−K (10) B N where KBi is the binding constant estimated in each simuB lation (i = 1,..., NB) and K is the mean value of the NB B

simulations. The K value is used as an estimate of the binding constant K. An assumption of the bootstrap method is that BIAS(K)B = 0. When this is not the case, a statistical test is used for verifying if the ratio between the value of the bias and its uncertainty is greater than the corresponding coverage factor k and therefore the bias does not differ significantly of zero. If the bias is too large to be ignored, the value of the binding constant could be corrected by the bias: B

K =K +BIAS(K)B

(11)

Since the jackknife method requires the computation of K only for the NJ data sets (NJ = n experimental data), it will be easier to compute it if N is less than 100 or 200 replicates used by the bootstrap method for the standard error estimation. However by looking only at the NJ jackknife samples, the jackknife method only uses limited information about the statistic K, and thus one might guess that the jackknife method is less efficient than the bootstrap method. In fact, it turns out that the jackknife method can be viewed as an approximation of the bootstrap one. The jackknife method often works well and provides a simpler and better approximation than the bootstrap for the estimation of standard errors and bias. This is only true if the statistics under study do not change drastically upon small changes in the data. However, when the number of experimental data is too small, the uncertainty obtained increases with regard to the bootstrap method. Coverage interval on the estimates The value of the binding constant must be accompanied by its corresponding uncertainty, K± k·uK, where uK is the standard uncertainty associated with the estimated value of the binding constant and k is the value of the coverage constant [21, 33], which depends on the effective degrees of freedom calculated from the Welch-Satterthwaite expression. This uncertainty is the maximum standard error estimated for each methodology.

Experimental Chemicals and reagents

The jackknife method There are some computational alternatives for estimating the standard deviation and the bias. The best known is the jackknife methodology [22]. It has been applied, for example, to identify outliers in multivariate calibration [30], in the determination of the thermodynamic dissociation constant by regression analysis of potentiometric data [31] and in pharmacokinetics determinations by HPLC [32]. In several papers [27, 29], the two cited techniques, bootstrap and jackknife, have been compared. Some authors [29]have even coupled these techniques with satisfactory results. In the jackknife method, eliminating each object in turn from the data set, NJ new data sets are created. In analogy with the bootstrap method, the statistic of interest is calculated for each new data set, yielding NJ values for Ki that are the estimated value from the sample with the observation i removed from the rest. From these results, estimates of bias and standard error can be obtained: 2 NJ −1 Y
J J s2 (K)J = Ki −K BIAS(K)J = (NJ −1)(K−K ) J N (12) J N being the number of experimental data.

2,4-Dichlorophenoxyacetic acid (2,4-D) was obtained from Merck. Methanol was obtained from Acros Organics. HP-β-CD, with an average molecular substitution of 0.8, was obtained from Aldrich. The buffer solutions at pH 2, 5 and 8 were obtained from Fluka. Deionized water was used for the experimental work. Apparatus Fluorescence measurements were performed on a Kontron SFM-25 spectrofluorometer, using a Kontron SFM-25 data control and acquisition program. The high voltage level on the instrument lamp was 500 V. For the photodegradation reaction, a 200 W HBO Osram highpressure mercury lamp with an Oriel model 8500 power supply was used and a standard Hellma 1 cm2 quartz reaction cuvette was placed on an optical bench at 45 cm from the lamp. Statgraphics Plus 5.0 statistical software package was used for data treatment in the optimization. Excel 97 from Microsoft was used for the rest of the calculations. Procedure Standard stock solutions of 2,4-D (100 mg/l) and HP-β-CD (0.02 M) were prepared by dissolving the compound in distilled water and these solutions were protected against light with aluminium foil. The photolysis reaction was performed in a quartz cell by irradiating with UV light a 3-ml volume of dilute herbicide solution,

418 which was magnetically stirred at room temperature. The maximum intensity of fluorescence (F) was registered at a constant excitation (268 nm) and emission (296 nm) wavelength.

Results and discussion Experimental methodology The first step is to obtain the optimum experimental conditions (pH, irradiation time and HP-β-CD concentration) to find the maximum analytical response using photochemically-induced fluorescence as analytical technique. The experimental variables were optimized simultaneously using the response surface methodology (RSM) from sequential experimental Box-Behnken [34] designs. These designs are rotatable and present a uniform distribution of points over the experimental domain. Figure 1 shows two different representations of the three variables Box-Behnken designs, which are equivalent. They also present other advantages in relation to others designs: a) They are more efficient, b) they are easier to arrange and interpret and c) the initial dimensions can be contracted or expanded on the experimental domain. The Box-Behnken designs can be used to re-optimize [35] analytical methods in case there are factors generating variability, which are not known or they can not be easily controlled. Once the optimum conditions have been obtained, the modified B-H method is applied in order to obtain the stoichiometry of the inclusion complex. Then, the binding constant and its uncertainty are obtained using the parametric and non-parametric methods and the results are compared. Optimization Box-Behnken designs have been used to obtained the maximum photochemically-induced fluorescence intensity for 2,4-D (4 mg/l) in the inclusion complex formed with HP-β-CD. The experimental variables influencing

Fig. 1 Tridimensional plots of a three-variable Box-Behnken design

Fig. 2 Design used in the optimization of the HP-β-CD concentration, pH and the irradiation time for the inclusion complex (2,4-D concentration, CS =4 mg/l). (A) Design 1; (B) Design 2; (1) Maximum obtained from design 1, (2) Maximum obtained from design 2

the analytical response are: the irradiation time (T), the HP-β-CD concentration (CCD), and the pH (pH). For the optimization, we have applied two designs (Figure 2), for which the coded values have been used for the studied variables, these are adequate for the normalization of the experimental domain. The first design is centered on point (0,0,0), which corresponds to 0.03 M in HP-β-CD, an irradiation time of 20 min and a pH value of 5. Each variation of one unit in the coded value means a change of ±0.02 M, 10 min and 3 units in the HP-β-CD concentration, irradiation time and pH, respectively. Table 1 shows the proposed BoxTable 1 Box–Behnken design for the fluorescence intensity (F) optimization for the HP–β–CD:2,4–D inclusion complex (CS = 4 mg/l). (Design 1) Exp. No.

CCD (mol/l)

pH

Irra. Time (min)

F

Predicted values

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.01 (–1) 0.05 (1) 0.01 (–1) 0.05 (1) 0.01 (–1) 0.05 (1) 0.01 (–1) 0.05 (1) 0.03 (0) 0.03 (0) 0.03 (0) 0.03 (0) 0.03 (0) 0.03 (0) 0.03 (0)

2 (–1) 2 (–1) 8 (1) 8 (1) 5 (0) 5 (0) 5 (0) 5 (0) 2 (–1) 8 (1) 2 (–1) 8 (1) 5 (0) 5 (0) 5 (0)

20 (0) 20 (0) 20 (0) 20 (0) 10 (–1) 10 (–1) 30 (1) 30 (1) 10 (–1) 10 (–1) 30 (1) 30 (1) 20 (0) 20 (0) 20 (0)

20.6 48.8 28.0 54.1 15.5 49.5 25.8 61.9 29.1 34.8 42.7 47.7 45.7 44.0 46.9

19.0 50.9 25.9 55.7 17.0 47.3 29.0 60.4 29.2 35.4 42.1 47.6 45.5 45.5 45.5

The code values for each experiment are given in brackets

419 Fig. 3 Response surfaces for estimated response function for the fluorescence intensity optimization to 2,4-D (CS =4 mg/l) in: A design 1 with coded value for pH = 0, B design 1 with coded value for time =0, C design 2 with coded value for pH = 0 and D design 2 with coded value for time = 0

Behnken design and the experimental results obtained for the photochemically-induced fluorescence intensity. The function that fits the experimental values is: F = 45.33 + 15.42 · CCD +2.93 · pH + 6.27 · T −3.90 · C2CD −3.75 · pH2 −3.20 · T2 −0.53 · CCD ·pH +0.27 · CCD ·T − 0.17 · pH · T

that illustrates the relation between the fluorescence intensity (F) and HP-β-CD (CCD), pH and irradiation time (T). Table 2 Box–Behnken design for the fluorescence intensity (F) optimization for the HP–β–CD:2,4–D inclusion complex (CS= 4 mg/l). (design 2) Exp. No.

CCD (mol/l)

pH

Irra. Time (min)

F

Predicted values

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.05 (1) 0.09 (3) 0.05 (1) 0.09 (3) 0.05 (1) 0.09 (3) 0.05 (1) 0.09 (3) 0.07 (2) 0.07 (2) 0.07 (2) 0.07 (2) 0.07 (2) 0.07 (2) 0.07 (2)

2 (–1) 2 (–1) 8 (1) 8 (1) 5 (0) 5 (0) 5 (0) 5 (0) 2 (–1) 8 (1) 2 (–1) 8 (1) 5 (0) 5 (0) 5 (0)

30 (1) 30 (1) 30 (1) 30 (1) 20 (0) 20 (0) 40 (2) 40 (2) 20 (0) 20 (0) 40 (2) 40 (2) 30 (1) 30 (1) 30 (1)

54.8 60.9 51.4 60.8 55.0 65.4 57.2 65.1 55.3 53.9 59.8 60.2 66.3 67.6 67.0

54.1 60.9 51.4 61.5 54.2 63.9 58.7 65.9 56.7 54.7 59.0 58.8 67.0 67.0 67.0

The code values for each experiment are given in brackets

There is a good agreement between the F values predicted by the theoretical model and the experimental values (coefficient of determination R2 =98.6% and lack-of-fit test P-value =18.6%). The effects and significance of the variables are shown in the first Table of ANOVA (design 1) in the Electronic Supplementary Material. The application of Lagrange’s criterion [36], to the function, indicates the presence of a maximum (F”(CCD, CCD) =–7.81, ∆(CCD, pH) = 58.35, H(CCD, pH, T) =–373.08) (Figure 3A and B), which corresponds to the HP-β-CD concentration of 0.07 M, pH = 5.7 and an irradiation time of 30.6 min. Nevertheless, as the optimum obtained is not localized within the experimental region covered by the design, it is convenient to perform a fresh set of experiments, creating another design around the estimated optimum in order to confirm the position. Table 2 shows the design 2 and the experimental results obtained. The function adjusted to the new experimental values is: F = 39.46 + 18.08 · CCD −2.66 · pH+8.80 · T −3.31 · C2CD −6.68 · pH2 −2.98 · T2 +0.82 · CCD ·pH −0.62 · CCD ·T + 0.45 · pH · T

Once again we obtained a good agreement between the predicted and the experimental values of intensity (coefficient of determination R2 =94.9% and lack-of-fit test P-value =10.3%). The effects and significance of the variables are shown in the second Table of ANOVA (design 2) in the Electronic Supplementary Material. The application of Lagrange’s criterion (F”(CCD, CCD) =–6.62, ∆(CCD, pH) =87.76, H(CCD, pH, T) =–517.5) indicates the presence of a maximum, as can be seen in Figure

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3C and 3D, which corresponds to the concentration of 0.08 M in HP-β-CD, pH = 5.0 and an irradiation time of 32 min. The optimum is included in the experimental domain and these conditions will be used in further experiences. PIF spectral properties In aqueous solutions, HP-β-CD presents a photochemically-induced fluorescence signal, but not emission is observed for 2,4-D due to quenching effects. However, the fluorescence signal of 2,4-D in the presence of HP-β-CD is enhanced due to the ability of the CD to form an inclusion complex (Figure 4). The emission wavelength and the fluorescence signal of the complex formed vary when the [HP-β-CD]/[2,4-D] molar ratio is modified. Thus, the increase in the HP-β-CD concentration produces two dif-

Fig. 4 Comparison of HP-β-CD:2,4-D complex and HP-β-CD PIF emission spectra at λexc =268 nm; irradiation time =32 min. A 4 mg/l of 2,4-D and 0.07 M of HP-β-CD, B 0.07 M of HP-β-CD and C difference spectrum

ferent effects: an enhancement of the fluorescence intensity and a progressive red-shift of the emission peak (Figure 5). Due to the overlap of the emission spectra of the HP-βCD and of the inclusion complex (Figure 4), it is necessary to subtract the emission spectrum of the HP-β-CD from the emission spectrum of the inclusion complex for analytical purposes. There is no need for a further correction of the substrate fluorescence because this is negligible at working conditions. The maximum of the fluorescence intensity of the difference spectrum is used to determine the stoichiometry and the complex binding constant. Determination of the stoichiometry The stoichiometry was determined by measuring the fluorescence intensity of the herbicide in presence of HP-β-CD,

Fig. 6 Double reciprocal plot for HP-β-CD:2,4-D complex. A linear relationship (●) was obtained when the data are plotted assuming a 1:1 stoichiometry and a downward concave curve (▲) when the data are plotted assuming a 2:1 stoichiometry

Table 3 Experimental values, predicted responses, residuals and regression parameters for the HPβCD:2,4-D inclusion complex. (CS =4 mg/l)

Fig. 5 Evolution of PIF emission spectra of 2,4-D (CS =4 mg/l) with different HP-β-CD concentrations in 50/50 v/v methanol/ pH 5 mixture. (λexc =268 nm, irradiation time =32 min)

CCD (mol/l)

Fluorescence Intensity

Responsesa

Predicted responses

Residuals

0.01 0.02 0.03 0.04 0.05 0.06 0.07

14.7 21.5 25.1 27.6 29.2 30.6 31.0

6.80×10–4 9.30×10–4 1.19×10–3 1.45×10–3 1.71×10–3 1.96×10–3 2.26×10–3

6.70×10–4 9.30×10–4 1.19×10–3 1.45×10–3 1.72×10–3 1.98×10–3 2.24×10–3

8.49×10–6 –2.68×10–6 1.18×10–6 –5.89×10–6 –3.97×10–6 –1.66×10–5 1.95×10–5

Regression parameters a: 4.11×10–4 b: 2.61×10–2 sres: 1.26×10–5 sa: 0.11×10–4 sb: 2.37×10–4 R2: 0.9996 aResponses

= CCD / Fluorescence Intensity

421 Fig. 7 Distribution of 2000 binding constants from RC Bootstrap samples, 2000 binding constants from RR Bootstrap samples, 7 binding constant from jackknife samples and 14000 binding constant from bootstrap after jackknife (B after J) and jackknife after bootstrap (J after B) samples

fixing the 2,4-D concentration at CS =4 mg/l, the optimum irradiation time at 32 min and the wavelengths of maximum excitation and emission at 268 and 296 nm, respectively. The concentration range of the HP-β-CD studied was from 0.01 to 0.07 mol/l. The stoichiometry between HP-β-CD and 2,4-D was estimated using the B-H method. A typical double reciprocal plot for this complex is showed in Figure 6. A linear relationship is obtained (for the full range of tested CD concentrations) when 1/∆F is plotted against 1/CCD, indicating that the stoichiometry of

the complex is 1:1. In contrast, a curvature is obtained when these data were fitting to a 2:1 complex. Calculation of the binding constant and estimation of its uncertainty Fieller’s theorem. The modified B-H method (Eq. 3) is applied to obtain the regression parameters (Table 3) from the

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experimental data. The binding constant (K =63.59 l/mol) is calculated from the ratio of the regression parameters. To estimate the uncertainty associated to the binding constant, the parameter g is calculated from equation 5 (g = 0.0044). Since g is smaller than 0.05 it is possible to apply the reduced expression (equation 7) for estimating the uncertainty from the variance associated to the binding constant. The interval obtained by means of this approach is (d.f.= 5, k = 2.65): K = 63.58 ± 5.77 l/mol

RC Bootstrap method. The new data sets (named RC bootstrap samples) are created from the original data set by sampling with replacement. Two thousand (NB) bootstrap samples of seven experimental data are created and the regression parameters for each new set are calculated. Figure 7 shows the resulting histogram for the data sets. In this case, the coverage interval obtained by means of RC Bootstrap is (d.f.>50, k = 2): K = 63.67 ± 5.26 l/mol

RR Bootstrap method. Firstly, it is necessary to estimate the random errors associated to each experimental data. The regression coefficients obtained from the modified B-H method are used to calculate the predicted response for this model and the residual values associated to each experimental data (Table 3). These residuals are used to make new bootstrap data sets, simulating repetitions of the experiment, and with the analysis of the data to calculate the statistical variation of the ratio. Simulated values aB, bB and KB of the estimated coefficients are computed from the bootstrap data[(CCD )i , YBi ] . This simulation process is performed a high number of times (N = 2000). Figure 7 shows the resulting histogram for the data sets. This distribution can be seen as an approximation to the true distribution of the estimate, and therefore, statistics of interest such as the bias and the standard error (equation 10) can be derived from repeated sets of the experimental results. In this way, the coverage interval obtained by means of this method is (d.f.>50, k = 2): K = 63.66 ± 3.73 l/mol

Jackknife method. This method is based on the effect of the variability introduced in the estimated parameters by removing each experimental point, one at a time, from the original data set. The statistics of interest is calculated for each new data set, yielding N values for Ki, that is the estimated value from the sample when the observation i removed. From the results, estimates of the bias and the standard error (equation 12) can be obtained and the binding constant and its uncertainty are (d.f.= 5, k = 2.65): K = 63.58 ± 7.62 l/mol

Bootstrap after Jackknife. This strategy consists on obtaining, in the first time, by means of the jackknife

Table 4 Results from binding constant and its uncertainty by different methods METHOD

K

sK

BIAS (Ka)

Fieller’s theorem RC Bootstrap RR Bootstrap Jackknife Bootstrap after Jackknife Jackknife after Bootstrap

63.58a 63.67 63.66 63.58 63.61 63.61

2.18 2.63 1.87 3.03 2.33 2.04

0 –0.08 –0.07 0.05 0.03 0.02

aThis value is calculated from the modified Benesi–Hildelbrand method

method (leave one out) the experimental data. The modified B-H method is applied to obtain the regression coefficients and the residual values to each experimental data subset. Next, the RR Bootstrap method (N = 2000) is applied to each one of the regressions obtained from jackknife, increasing the number of the binding constant estimates. Figure 7 shows the resulting histogram for data set applying the bootstrap after jackknife technique. Finally, from the equations corresponding to the bootstrap method is possible to obtain the binding constant and its uncertainty (d.f.>50, k = 2): K = 63.61 ± 4.66 l/mol

Jackknife after Bootstrap. On the contrary of the previous case, the regression coefficients and the residual values to each experimental data are obtained from the modified B-H method. Then, the RR bootstrap method (N = 2000) is applied and the jackknife method is applied on each the simulated regressions. The number of estimated binding constant is substantially increased. Figure 7 shows the resulting histogram for the data set applying the jackknife after the bootstrap method. In this case, the result is (d.f.>50,k = 2): K = 63.61 ± 4.08 l/mol

Table 4 shows the coverage intervals on the estimated binding constants from the different applied methodologies. The results obtained by the six methods are not significantly different and the different calculated intervals are overlapped. We can conclude that all the methods applied in this work are valid and can be used for a rigorous estimation of binding constants and its associated uncertainty.

Conclusion Computer intensive methods, such as bootstrap and jackknife, are recognized as powerful tools in the evaluation of experimental data in complicated situations. Although the calculations appear to be very extensive, several high quality statistics packages can be applied for these purposes. In this paper, six methods (Fieller’s theorem, RC bootstrap, RR bootstrap, jackknife, bootstrap after jack-

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knife, and jackknife after bootstrap methods) have been applied to the simple reciprocal regression obtained for determining the binding constant for the HP-β-CD:2,4-D inclusion complex. It is also important to note that all the methods provide similar binding constant values but the smallest uncertainty has been obtained by the RR bootstrap method which should be considered as the most appropriate method in terms of accuracy in the estimated value.

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