# Adsorption between Quercetin Derivatives and β-Glucan Studied with a Novel Approach to Modeling Adsorption Isotherms

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Chemicals

#### 2.2. Spectrophotometric Method for Total Polyphenols (Folin-Ciocalteu Method)

_{2}CO

_{3}solution (200 g/L) were added in a glass tube. This solution was homogenized in vortex mixer and incubated at 40 °C for 30 min (IN 30, Memmert, Schwabach, Germany). Afterwards, the absorbance was measured at 765 nm (UV-Vis spectrophotometer, Shimadzu UV-1280, Kyoto, Japan) against a blank solution containing distilled water instead of quercetin derivative. The calibration equation, limit of detection (LOD) and limit of quantification (LOQ) were determined. Samples for quercetin derivative calibration curves were prepared in doublets and each was measured twice. This method was used for the determination of quercetin derivatives in the adsorption experiment.

#### 2.3. Adsorption of Quercetin Derivatives onto β-Glucan

_{blank}) was determined in the filtered solutions by applying the Folin-Ciocalteu method. The adsorption capacity q

_{e}(mg/mg of β-glucan) was calculated according to Equation (1).

_{blank}represents the concentration of quercetin derivatives in the blank experiment (mg/L), c is the concentration of quercetin derivatives after adsorption (mg/L), V

_{m}is the volume of the model solution (L). γ

_{a}is the mass concentration of β-glucan in the model solution (mg/L). The c

_{e}, or the concentration of un-adsorbed polyphenols (concentration in equilibrium), was calculated according to Equation (2) and expressed in mg:

_{o}is the initial polyphenol concentration in the reaction solution. In the first experiment, the initial concentration of quercetin derivatives was 25 mg/L. In the second experiment, the adsorption of quercetin derivatives was measured for additional concentrations 50, 75, 100, and 150 mg/L in the same experimental conditions, for 16 h, to get the data to be analyzed with isotherm equations.

#### 2.4. The Conservation of Mass

_{e}and c

_{e}will be presented in q

_{e}vs c

_{e}diagrams and then modeled with adsorption isotherms. For these diagrams, it is important to explain the implications of the conservation of mass. The expressions (1) and (2) arise as expressions of the conservation of mass: namely, the initial concentration ${c}_{0}$ is the sum of the amount adsorbed $\left({c}_{blank}-c\right)$ (providing ${q}_{e}$) and the amount not adsorbed $\left[{c}_{0}-\left({c}_{blank}-c\right)\right]$ (providing ${c}_{e}$). In other words, for initial c

_{o}, the higher the amount of adsorbed compound is (q

_{e}), the lower is the amount of un-adsorbed compound (c

_{e}). And vice versa. And their sum must always be the c

_{o}. The consequence of this mass constraint is that, for each initial concentration ${c}_{0}$, the data points (${c}_{e},{q}_{e})$ lay exactly on a line of slope -1/γ

_{a}V

_{m}in q

_{e}vs. c

_{e}diagram. This is a slope of –1, (that is, a line of slope –45 degrees) if we were to use the units of mass (in mg) for both. In other words, for each initial concentration, the data lay on a diagonal line, and not the vertical line. This fact is important in further explanation of the novel methodology for modeling the adsorption data.

#### 2.5. Adsorption Isotherms and Data Modeling

_{e}(mg) vs. q

_{e}(mg/mg β-glucan)) were modeled with Langmuir, Dubinin-Radushkevich and Hill adsorption isotherms [18,19,26,27,28,29] using non-linear regression, improved non-linear regression, and linear regression.

#### 2.5.1. Non-Linear Regression

_{L}is the Langmuir equilibrium constant of adsorption (1/mg) or apparent affinity constant, q

_{m}is the theoretical maximum adsorption capacity of β-glucan (mg/mg), q

_{s}is the theoretical isotherm saturation capacity (mg/mg), β is a constant related to the adsorption capacity (mol

^{2}/J

^{2}), ε is Polanyi potential (J/mol), R is the gas constant (8.314 J/mol K), T is the temperature (K), E is the adsorption mean free energy (J/mol), c

_{s}is the theoretical saturation concentration or solubility (mg), n

_{H}is the Hill cooperativity coefficient of the binding interaction, and K

_{D}is the Hill constant (mg)

^{nH}.

_{m}and K

_{L}. Dubinin Radushkevich is a three-parameter isotherm and the parameters that could be determined are q

_{s}, β, and c

_{s}. The c

_{s}represents the theoretical saturation capacity, so it should be large than the largest observed c

_{e}. A convenient constraint value for these data is that it not be smaller than 0.1 mg. With this constraint the non-linear least squares solutions occurred at this value. An equivalent result is obtained by fixing the c

_{s}value at the level of 0.1 mg. A least squares value of c

_{s}near 0.1 mg occurred automatically for the quercetine-3-rhamoside data, so no constraint was found to be needed in that case. Along with the c

_{s}value, we determined q

_{s}and β. From β, the energy E was calculated. Altogether, the reported parameters of the Dubinin-Radushkevich isotherm are c

_{s}, q

_{s}and E. The parameters for the three-parameter Hills model are q

_{m}, n

_{H}and K

_{D}. An approximate least squares values of the Hills constant K

_{D}was found at the level 0.0001 for quercetin-3-galactoside and quercetin-3-glucoside (smaller K

_{D}values that show potential insignificant improvement were numerically unstable, so we chose the indicated level). The stable least squares value of K

_{D}was 0.033 for quercetin-3-rhamnoside. Along with K

_{D,}we determined and reported the two additional Hill parameters q

_{m}and n

_{H}.

#### 2.5.2. Improved Non-Linear Regression

_{e}vs. q

_{e}) were analyzed also with improved non-linear regression. To explain the potential need for this improvement let’s contrast the situation with that of traditional least squares curve fitting. Traditional non-linear least square is appropriate for response models in which the y values are measured values of a parameterized function of the corresponding input value x. For each set value of x, the response y is measured multiple times, with some potential variability around the unknown function value. The values for multiple measurements of y at the input value x follow a vertical line in the y vs x diagram. For the curve determination with non-linear regression, the square of the vertical residual between the measured y and the functional value y is summed, and minimized as a function of the parameters. In these models, the error is independent of the input value x.

_{e}vs. c

_{e}diagrams. However, in fact, both q

_{e}and c

_{e}are measured/calculated as responses to the initial concentration c

_{o}. If we have multiple measurement for one initial c

_{o}, the line of multiple q

_{e}in the diagram will be the diagonal line due to the conservation of mass, as explained earlier. It is this constraint from the conservation of mass relationship, along with the structure of variability it induces (perfectly negative correlation between c

_{e}and q

_{e}), that can necessitate the need for improvements on the traditional least squares procedure. If we fit the curve through those data that are shown in the diagonal line, these is a unique crossing point of the diagonal with the curve. For precise curve determination that respects the mass constraint, the appropriate square of residuals between measured q

_{e}and curve q

_{e}at that crossing point should be determined. Minimizing the sum of squares of this error is what is sought to be achieved by the improved non-linear adsorption fitting program. Properties of this improved regression are discussed further in the results section.

#### 2.5.3. Linear Regression

_{e}values as here indicated.

_{e}vs. c

_{e}/q

_{e}was created, and q

_{m}and K

_{L}were determined from the slope (1/q

_{m}) and intercept (1/K

_{L}q

_{m}) and reported. In the Dubinin-Radushkevich isotherm, ε was calculated according to equation 5 with value c

_{s}of 0.1 mg. Diagram ε

^{2}vs. lnq

_{e}was created, and β and q

_{s}were calculated from slope (β) and intercept (lnq

_{s}). For Hill’s isotherm, Equation (11) is an approximate linearized form valid for small K

_{D.}A diagram of lnc

_{e}vs. lnq

_{e}/(q

_{m}-q

_{e}) was created. The q

_{m}value was the q

_{m}constant from the Langmuir model. From the slope (n

_{H}) and intercept (lnK

_{D}), parameters n

_{H}and K

_{D}were determined and reported.

#### 2.6. Statistical Analysis

_{e}vs. q

_{e}) were modeled with adsorption isotherms using non-linear and linear regressions. Non-linear regression was performed using the MS Excel software add-inn called Solver (MS Excel, Redmond, Washington, USA). Linear regression was also done in MS Excel. To improve the modeling and to lower the error of models, a new form of improved non-linear regression was developed in the R programming language, and it was also applied to these data. This program confirmed the traditional non-linear least squares fits and provided the potential improved fits. The standard error of regression (se) of non-linear, linear, and improved non-linear least square regression was calculated according to the equation 12 where q

_{e,meas}and q

_{e,model}are (original or transformed) measured adsorption capacities and adsorption capacities calculated by the model, respectively, n is the total number of data points and a is the number of parameters of the model.

_{e}(in standard non-linear regression) or at the special curve point where it crosses the diagonal (in improved non-linear regression).

## 3. Results

^{2}(0.9932–0.9985), all quercetin derivative standards showed linear calibration curves in the studied range, with reasonably low LOD and LOQ.

_{e}, mg/mg of β-glucan) for the initial concentration of quercetin derivatives 25 mg/L. All three quercetin derivatives adsorbed in similar amounts (0.30, 0.25 and 0.21 mg/mg, for quercetin-3-rhamnoside, quercetin-3-glucoside and quercetin-3-galactoside, respectively). The amounts of adsorbed quercetin derivatives are similar to the amounts of various polyphenols adsorbed onto β-glucan, which are reported in earlier studies [16,17]. Namely, Gao et al. (2012a) [16] reported the adsorption of tea polyphenols onto β-glucan in the amount of 0.156 to 0.405 mg/mg β-glucan, which is similar to our results. They also studied the adsorption of epigallocatechin gallate under various initial concentrations and the maximum adsorption capacity with higher epigallocatechin gallate concentration was around 0.25 mg/mg β-glucan [16]. Wu et al. (2011) [17] studied the adsorption of tea polyphenols onto β-glucan, and the influence of different pH values, buffer concentration and temperature onto adsorption. The highest adsorption capacity of tea polyphenols under various influences of pH, temperature and buffer concentrations was 0.116 mg/mg β-glucan which is also similar to our results. Furthermore, adsorption of various polyphenols onto different adsorbents like cellulose [7,10,11], cell wall material [34,35,36], pectin, xyloglucan, starch and cellulose [37], and resin [38] was also investigated. Since the amount of adsorbed polyphenols depends on the initial polyphenol concentration, pH value, buffer, and temperature, it is difficult to compare the adsorption capacities in these studies, but it can be seen that the adsorbed amount from those earlier studies [7,10,11,34,35,36,38] was in the range of the results of this study. The adsorption of apple polyphenols to apple cell walls was in the range of 0.14 and 0.58 mg/mg of cell walls, which is also similar to our study [39].

_{e}vs. c

_{e}) for one polyphenol concentration, measured several times are visible in the diagonal line corresponding to the conservation of mass constraint as previously mentioned. The value of q

_{e}predicted by the model (q

_{e,model}) should follow that constraint and be predicted on the diagonal line. However, the standard non-linear regression does not predict q

_{e,model}on that diagonal. Instead it drops vertically to a point that does not satisfy the mass constraint. In doing so, the standard non-linear regression residual is always larger than the improved non-linear regression residuals, for adsorption curves that are increasing functions of c

_{e.}Moreover, as long as the adsorption curve is strictly increasing, use of the standard least squares will produce incorrect fitted maximum adsorption capacities (q

_{m}and q

_{s}) that remain incorrect by a constant factor, even in the limit of a large number of experimental observations (this is its statistical inconsistency for the correlated error setting of adsorption curve fitting). The same remains true even for linear curve fitting, if the observations occur along the diagonals specified by the initial polyphenol concentrations.

_{e,model}value at the intersection points of the curves and the diagonal lines of the data. Motivation for developing this improved way of non-linear modeling of adsorption data is not only that it gives smaller residuals, but also that it gives, for normal error distributions, a provably statistically efficient standard error asymptotically, superior to those of other consistent procedures. As an example of the novel non-linear regression consider Figure 2b. It shows the method in this study indeed produces the q

_{e,model}values at the intersections of the curve with the experimental data lines (the special curve points). These diagonal lines also can be seen in the confidence intervals of the model fits in Figure 2b.

_{e}were calculated by using models obtained with improved non-linear modeling and the results q

_{e,calc}are shown in Table 2. The q

_{e,calc}gave relatively good approximation of the measured q

_{e}.

_{m}) from Langmuir and Hill isotherm or (q

_{s}) from Dubinin-Radushkevich isotherm, we can suggest which derivatives adsorbed more onto β-glucan or which have more tendency to adsorb. Glucoside and rhamnoside of quercetin could theoretically adsorb in higher amount than quercetin-3-galactoside (Table 3), which is similar to experimental results (Table 2). Furthermore, according to the mean free energy of adsorption (E), the adsorption could be a physical process (since E < 8000 J/mol) with creation of physical bonds between quercetin derivatives and β-glucan, such as Van der Waals bonds or H bonds. This agrees with earlier studies where it was suggested that hydrophobic interactions, hydrogen bonds and van der Waals interactions are the driving forces of adsorption process of polyphenols with β-glucan [16,17,22,23]. Hydrogen bonds between quercetin derivatives and β-glucan could occur through their -OH groups [17] and after the formation of hydrogen bonds, van der Waals interactions could be formed because the distance between the polyphenol molecule and β-glucan rings becomes short [17]. The same type of bonding could have happened between quercetin derivatives and β-glucan. Furthermore, the mean free adsorption energy (E) was the highest for quercetin-3-galactoside (4327 J/mol), followed by glucoside (3623 J/mol) and rhamnoside (2854 J/mol). This suggests that the mean free energy of adsorption was higher for polyphenols which showed lower adsorption onto β-glucan (lower q

_{e}). Adsorption could be a cooperative process (n

_{H}> 1). When a molecule of quercetin derivative adsorbs to the surface of β-glucan, the affinity for other molecules changes.

## 4. Discussion

_{e}vs. c

_{e}) to isotherms often depends on the modeling methodology. Proper modeling can give lower error between experimental results (q

_{e}) and the fitted adsorption isotherms (q

_{e,model}). If results are fitted with lower errors, the q

_{e,model}values will be closer to experimental values and adsorption can be better interpreted. Quality fitting and interpretation of adsorption can be achieved by non-linear and linear regression. In some papers, the adsorption data were analyzed with non-linear regression [7,9,10,17,34,36,37] and in some with the use of linear regression [18]. Between those two types of modeling, non-linear has shown advantages, since linear fitting have already shown many constrictions [28]. That is why the use of non-linear fitting is more common [28]. A potential difficulty with the linearized forms is that the transformed forms of ${c}_{e}$ and ${q}_{e}$ do not retain the structure of variability of the original forms, and consequently there can be a loss of statistical efficiency with the use of the least squares procedure with the linearized forms. This can be observed in larger errors in the fitted parameters and the fitted isotherm curves. We investigated and compared both types of modeling, non-linear and linear. To improve the fitting and to get lower error and more reliable isotherms and their parameters, we also developed and used an improved non-linear modeling for the first time.

_{e,model}value is on the −45 degrees line corresponding to the conservation of mass. Indeed, the improved non-linear fitting predicts the q

_{e,model}value on this diagonal line and does give lower error (Table 3). Although the differences between errors obtained with improved non-linear fitting, traditional non-linear and linear were not so big, they can still point to more proper fitting of the novel methodology. So, the parameters of adsorption isotherms obtained by improved non-linear fitting could be used to describe the adsorption. In addition, the errors for Langmuir, Dubinin-Radushkevich, and Hill isotherms were similar (a little bit lower for Langmuir, followed by Dubinin-Radushkevich and Hill), but we chose to use the parameters of all three isotherms for adsorption interpretation. Although the interpretation of isotherms and their parameters is theoretical, it can still give some insight into the adsorption process.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**a**) Theory of non-linear regression of adsorption data with standard non-linear regression and novel improved non-linear regression. (

**b**) amplified figure of example of novel improved non-linear regression of Dubinin-Radushkevich isotherms (quercetin-3-galactoside adsorption onto β-glucan).

**Figure 3.**Langmuir, Dubinin-Radushkevich and Hill isotherms of quercetin derivatives adsorption onto β-glucan obtained with standard non-linear regression.

**Figure 4.**Langmuir, Dubinin-Radushkevich, and Hill isotherms of quercetin derivatives adsorption onto β-glucan obtained with novel improved non-linear regression (with reported q

_{e},

_{model}at the lowest c

_{e}value) (the confidence intervals are ±1 standard errors of the model fits).

**Figure 5.**Langmuir, Dubinin-Radushkevich and Hill isotherms of quercetin derivatives adsorption onto β-glucan obtained with linear regression.

**Table 1.**Linearity, limit of detection (LOD), limit of quantification (LOQ) of Folin-Ciocalteu spectrophotometric method for the determination of quercetin derivatives.

Quercetin Derivative | Range mg L ^{−1} | Equation | r^{2} | LOD mg L ^{−1} | LOQ mg L ^{−1} |
---|---|---|---|---|---|

quercetin-3-glucoside | 1–200 | Y = 0.001x + 0.0055 | 0.9972 | 1.17 | 3.54 |

quercetin-3-galactoside | 1–100 | Y = 0.001x + 0.0031 | 0.9932 | 0.22 | 0.67 |

quercetin-3-rhamnoside | 1–200 | Y = 0.0013x + 0.00003 | 0.9985 | 0.17 | 0.52 |

**Table 2.**Quercetin derivative adsorption capacities (q

_{e}) and adsorption capacities calculated from the adsorption isotherm model (q

_{e,calc}) (modeled with improved non-linear modeling).

Quercetin Derivative | Langmuir | Dubinin-Radushkevich | Hill | |
---|---|---|---|---|

q_{e} | q_{e(calc)} | q_{e(calc)} | q_{e(calc)} | |

(mg/mg) | (mg/mg) | (mg/mg) | (mg/mg) | |

quercetin-3-glucoside | 0.25 ± 0.05 | 0.35 | 0.30 | 0.25 |

quercetin-3-galactoside | 0.21 ± 0.19 | 0.30 | 0.28 | 0.24 |

quercetin-3-rhamnoside | 0.30 ± 0.07 | 0.40 | 0.27 | 0.35 |

**Table 3.**Parameters of adsorption isotherms of quercetin derivatives adsorption onto β-glucan obtained from non-linear * (standard and improved) and linear * equation of Langmuir’s, Dubinin-Radushkevich’s and Hill’s equations.

Quercetin-derivatives | Langmuir | Dubinin-Radushkevich | Hill | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

q_{m}(mg/mg) | K_{L}(1/mg) | se | q_{s}(mg/mg) | c_{s} (mg) | E (J/mol) | se | q_{m}(mg/mg) | n_{H} | K_{D}(mg) ^{n}_{H} | se | |

Nonlinear modeling | |||||||||||

quercetin-3-glucoside | 1.12 | 39.9 | 0.330 | 0.85 | 0.1 | 3721 | 0.339 | 0.82 | 2.23 | 0.000103 | 0.335 |

quercetin-3-galactoside | 0.66 | 73.0 | 0.324 | 0.57 | 0.1 | 4483 | 0.335 | 0.56 | 2.15 | 0.000091 | 0.334 |

quercetin-3-rhamnoside | 3.47 | 11.0 | 0.248 | 1.55 | 0.1 | 2909 | 0.246 | 2.67 | 1.18 | 0.0331 | 0.253 |

Improved nonlinear modeling | |||||||||||

quercetin-3-glucoside | 1.17 | 36.4 | 0.324 | 0.87 | 0.1 | 3623 | 0.318 | 0.84 | 2.26 | 0.000103 | 0.328 |

quercetin-3-galactoside | 0.69 | 64.3 | 0.322 | 0.58 | 0.1 | 4327 | 0.321 | 0.57 | 2.17 | 0.000091 | 0.331 |

quercetin-3-rhamnoside | 3.51 | 10.9 | 0.235 | 1.54 | 0.098 | 2854 | 0.232 | 2.69 | 1.19 | 0.0331 | 0.240 |

Linear modeling | |||||||||||

quercetin-3-glucoside | 0.75 | 36.0 | 0.410 | 0.74 | 0.1 | 3536 | 0.357 | 0.41 | 0.331 | 0.499 | |

quercetin-3-galactoside | 0.52 | 24.8 | 0.404 | 0.50 | 0.1 | 3162 | 0.360 | 0.15 | 0.696 | 0.439 | |

quercetin-3-rhamnoside | 3.72 | 9.3 | 0.156 | 1.51 | 0.1 | 2887 | 0.256 | 1.24 | 0.049 | 0.271 |

_{m}is the apparent maximum adsorption capacity of β-glucan (mg of polyphenols per mg of β-glucan), K

_{L}is the Langmuir equilibrium constant of adsorption (1/mg), apparent affinity constant, q

_{s}is theoretical saturation capacity of β-glucan (mg/mg), c

_{S}is the theoretical saturation concentration or solubility (mg), E is sorption mean free energy of adsorption (J/mol), n

_{H}is Hill cooperativity coefficient of the binding interaction, K

_{D}is Hills constant (mg)

^{nH}and se is the standard error. * standard non-linear and linear modeling performed in excel, improved non-linear performed in R program.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jakobek, L.; Matić, P.; Kraljević, Š.; Ukić, Š.; Benšić, M.; Barron, A.R. Adsorption between Quercetin Derivatives and β-Glucan Studied with a Novel Approach to Modeling Adsorption Isotherms. *Appl. Sci.* **2020**, *10*, 1637.
https://doi.org/10.3390/app10051637

**AMA Style**

Jakobek L, Matić P, Kraljević Š, Ukić Š, Benšić M, Barron AR. Adsorption between Quercetin Derivatives and β-Glucan Studied with a Novel Approach to Modeling Adsorption Isotherms. *Applied Sciences*. 2020; 10(5):1637.
https://doi.org/10.3390/app10051637

**Chicago/Turabian Style**

Jakobek, Lidija, Petra Matić, Šima Kraljević, Šime Ukić, Mirta Benšić, and Andrew R. Barron. 2020. "Adsorption between Quercetin Derivatives and β-Glucan Studied with a Novel Approach to Modeling Adsorption Isotherms" *Applied Sciences* 10, no. 5: 1637.
https://doi.org/10.3390/app10051637