5. Methods for computation of binding constants: the case of EO-CD association

In this section a rather simple and straight overview of the most used model equations to compute binding constants for host-guest association is provided. To do so, EO will be used as the guest compound.

As discussed before, the phase-solubility plots are a widely applied method to get knowhow on the improvement of the EO solubility driven by complexation and also for computation of the EO-CD association constant. Assuming that the most common type of EO/CD complex has a 1:1 stoichiometry, the corresponding reaction and equilibrium (binding) constant, K1, can be written as

$$\text{C} \text{CD} + \text{G} \overset{\text{K}\_1}{\underset{\text{CH}}{\rightleftharpoons}} \text{CD} - \text{G} \tag{1}$$

highly dependent on the intercept accuracy [96]. In order to overcome this drawback, the authors have established the concept of "complexation efficiency" (CE) which can be obtained

The application of Eq. (7) is useful but limited by the dependence on S<sup>0</sup> and to 1:1 complexes.

Kn

where n is a stoichiometry coefficient and Kn is the corresponding binding constant. Thus,

By measuring a physical parameter, known as ΔA, directly related with the formation of the complex (CDn � G), and performing the experiment in such a way that [CD]T>>[CD � G],

However, it should be pointed out that nowadays, with the available software, such as Origin® and MatLab®, there is no need to linearize Eq. (12) once such methodology brings some restric-

It should be stressed that a key point in all these procedures is the accurate previous knowledge of the stoichiometry of complexation, but this is not always a simple task. The most

1 Kn½ � S <sup>T</sup> ½ � CD <sup>T</sup>

Ka <sup>¼</sup> <sup>Δ</sup><sup>A</sup> ½ � CD <sup>T</sup>

<sup>¼</sup> ð Þ ST � <sup>S</sup><sup>0</sup> <sup>=</sup>½ � CD <sup>T</sup> 1 � ð Þ ST � S<sup>0</sup> =½ � CD <sup>T</sup>

CDn � G (8)

Interactions between Bio-Based Compounds and Cyclodextrins

http://dx.doi.org/10.5772/intechopen.73531

½ � G <sup>f</sup> ¼ ½ � G <sup>T</sup> � ½ � CDn � G (9)

½ � CD <sup>f</sup> ¼ ½ � CD <sup>T</sup> � n CD ½ � � G (10)

½ � CD <sup>T</sup> � n CD ½ � � <sup>G</sup> <sup>n</sup> ½ � <sup>S</sup> <sup>T</sup> � ½ � CD � <sup>G</sup> (11)

<sup>n</sup> ½ � <sup>S</sup> <sup>T</sup> � <sup>Δ</sup><sup>A</sup> (12)

<sup>n</sup> (13)

(7)

81

independently of S0, according to the following relation:

Let us now assume the following reaction:

from the conservation of mass equations:

the binding constant equation can be written as

and

or its linear form

tions to the computation of K and n.

CE <sup>¼</sup> ½ � CD � <sup>G</sup> ½ � CD <sup>f</sup>

where the term ((ST � S0)/[CD]T) represents the slope of the phase-solubility profile.

nCD þ G ⇄

Ka <sup>¼</sup> ½ � CDn � <sup>G</sup>

Eq. (11) takes the form of the so-called "Benesi-Hildebrand" equation [97]:

1 <sup>Δ</sup><sup>A</sup> <sup>¼</sup> <sup>1</sup> ½ � S <sup>T</sup> þ

$$K\_1 = [\mathbf{CD} - \mathbf{G}] / \left( [\mathbf{CD}]\_f [\mathbf{G}]\_f \right) \tag{2}$$

where G represents the guest (here the EO) and [CD]<sup>f</sup> and [G]<sup>f</sup> are the concentrations of uncomplexed (free) species in the system. Assuming that the change in the aqueous solubility of the EO (ΔS) is only due to the formation of the complex, we can write

$$
\Delta \mathcal{S} = \mathcal{S}\_{\mathcal{T}} - \mathcal{S}\_0 = [\mathcal{C}D - \mathcal{G}] \tag{3}
$$

where ST is the measurable total solubility and S<sup>0</sup> is the solubility of the EO in water in the absence of CD (i.e., the intrinsic solubility). Thus, it follows that

$$\left[\mathbf{G}\right]\_f = \mathbf{S}\_0 \tag{4}$$

$$\left[\mathbb{C}D\right]\_f = \left[\mathbb{C}D\right]\_T - \left(\mathbb{S}\_T - \mathbb{S}\_0\right) \tag{5}$$

where [CD]<sup>T</sup> is the total concentration of CD in the solution.

Substituting Eqs. (3)–(5) in Eq. (2) and after algebraic manipulation, we obtain

$$\mathbf{S}\_{T} = \mathbf{S}\_{0} + \frac{\mathbf{K}\_{1}\mathbf{S}\_{0}}{\mathbf{1} + \mathbf{K}\_{1}\mathbf{S}\_{0}}[\mathbf{C}\mathbf{D}]\_{T} \tag{6}$$

Fitting Eq. (6) to experimental data of S<sup>T</sup> = f([CD]T) allows the calculation of the intercept (S0) and the association constant, K1. As discussed by Loftsson et al., the determination of K is highly dependent on the intercept accuracy [96]. In order to overcome this drawback, the authors have established the concept of "complexation efficiency" (CE) which can be obtained independently of S0, according to the following relation:

$$\text{CE} = \frac{[\text{CD} - \text{G}]}{[\text{CD}]\_f} = \frac{(\text{S}\_T - \text{S}\_0) / [\text{CD}]\_T}{1 - (\text{S}\_T - \text{S}\_0) / [\text{CD}]\_T} \tag{7}$$

where the term ((ST � S0)/[CD]T) represents the slope of the phase-solubility profile. The application of Eq. (7) is useful but limited by the dependence on S<sup>0</sup> and to 1:1 complexes. Let us now assume the following reaction:

$$m\text{CD} + G \stackrel{\text{K}\_n}{\rightleftharpoons} \text{CD}\_n - G \tag{8}$$

where n is a stoichiometry coefficient and Kn is the corresponding binding constant. Thus, from the conservation of mass equations:

$$\mathbf{[G]}\_f = [\mathbf{G}]\_T - [\mathbf{C}D\_n - \mathbf{G}] \tag{9}$$

and

β-pinene; their solubility in water was improved more than one order of magnitude [94]. It is worth noticing that, recently, a technique based on the total organic carbon determination has been reported and validated to follow the solubility improvement of EOs when increasing the

5. Methods for computation of binding constants: the case of EO-CD

In this section a rather simple and straight overview of the most used model equations to compute binding constants for host-guest association is provided. To do so, EO will be used as

As discussed before, the phase-solubility plots are a widely applied method to get knowhow on the improvement of the EO solubility driven by complexation and also for computation of the EO-CD association constant. Assuming that the most common type of EO/CD complex has a 1:1 stoichiometry, the corresponding reaction and equilibrium (binding) constant, K1, can be

CD � G (1)

ΔS ¼ ST � S<sup>0</sup> ¼ ½ � CD � G (3)

½ � CD <sup>f</sup> ¼ ½ � CD <sup>T</sup> � ð Þ ST � S<sup>0</sup> (5)

½ � G <sup>f</sup> ¼ S<sup>0</sup> (4)

½ � CD <sup>T</sup> (6)

(2)

CD þ G ⇄ K1

of the EO (ΔS) is only due to the formation of the complex, we can write

absence of CD (i.e., the intrinsic solubility). Thus, it follows that

where [CD]<sup>T</sup> is the total concentration of CD in the solution.

Substituting Eqs. (3)–(5) in Eq. (2) and after algebraic manipulation, we obtain

ST ¼ S<sup>0</sup> þ

K<sup>1</sup> ¼ ½ � CD � G = ½ � CD <sup>f</sup> ½ � G <sup>f</sup>

where G represents the guest (here the EO) and [CD]<sup>f</sup> and [G]<sup>f</sup> are the concentrations of uncomplexed (free) species in the system. Assuming that the change in the aqueous solubility

where ST is the measurable total solubility and S<sup>0</sup> is the solubility of the EO in water in the

K1S<sup>0</sup> 1 þ K1S<sup>0</sup>

Fitting Eq. (6) to experimental data of S<sup>T</sup> = f([CD]T) allows the calculation of the intercept (S0) and the association constant, K1. As discussed by Loftsson et al., the determination of K is

concentration of CD [95].

80 Cyclodextrin - A Versatile Ingredient

association

written as

the guest compound.

$$\left[\mathbb{C}D\right]\_{f} = \left[\mathbb{C}D\right]\_{T} - n\left[\mathbb{C}D - \mathbb{G}\right] \tag{10}$$

the binding constant equation can be written as

$$K\_d = \frac{[\text{CD}\_n - \text{G}]}{\left( [\text{CD}]\_T - n[\text{CD} - \text{G}] \right)^n \left( [\text{S}]\_T - [\text{CD} - \text{G}] \right)} \tag{11}$$

By measuring a physical parameter, known as ΔA, directly related with the formation of the complex (CDn � G), and performing the experiment in such a way that [CD]T>>[CD � G], Eq. (11) takes the form of the so-called "Benesi-Hildebrand" equation [97]:

$$K\_a = \frac{\Delta A}{\left( [\![\![\!D]\!]\_T\right)^n \left( [\![S]\!]\_T - \Delta A \right)}\tag{12}$$

or its linear form

$$\frac{1}{\Delta A} = \frac{1}{[S]\_T} + \frac{1}{K\_n[S]\_T \left( [\mathbf{CD}]\_T \right)^n} \tag{13}$$

However, it should be pointed out that nowadays, with the available software, such as Origin® and MatLab®, there is no need to linearize Eq. (12) once such methodology brings some restrictions to the computation of K and n.

It should be stressed that a key point in all these procedures is the accurate previous knowledge of the stoichiometry of complexation, but this is not always a simple task. The most common method for the determination of stoichiometry is the method of continuous variation or the Job plot; the virtues and limitations of this method were recently reviewed [98], and thus it is not our intention to further discuss it in the present chapter.

Going back to the determination of the binding constants, the most accurate way to compute K is by using the first principles. Here, for the sake of simplicity, the 1:1 and 1:2 (G/CD) stoichiometric ratios will be focused. Additionally, these examples also correspond to the large majority of complexes formed between CDs and EOs. For more complex stoichiometries, the computational treatment of the resulting equations (not shown) is not straightforward as a consequence of multicollinearity [99]. Multicollinearity causes larger standard errors in the quantities calculated and lowers statistical significance of the results. In limiting cases, several local minima may be obtained by iteration; these correspond to noticeably different combinations of the quantities calculated and may be the reason why different K values are reported for the same host-guest systems.

Assuming that a 1:1 complex (CD-S) is formed, the binding constant (Eq. (2)) can be rewritten as

$$K\_1 = \frac{f}{(1-f)\left( [\mathbf{CD}]\_T - f[\mathbf{G}]\_T \right)}\tag{14}$$

<sup>Δ</sup>Aobs <sup>¼</sup> <sup>Δ</sup>ACD�<sup>G</sup> <sup>T</sup> <sup>þ</sup> <sup>1</sup> <sup>K</sup> <sup>=</sup> <sup>1</sup>

way that its value should be of the same order of magnitude than K<sup>1</sup>

K<sup>1</sup> and K2, and the corresponding mass balances are defined as

From the equilibrium constants and Eqs. (19) and (20), we can write

ation, it is possible to write Eq. (21) as a function of [CD], that is

½ � CD <sup>2</sup> <sup>þ</sup>

On the other hand, the free CD concentration is given by

� ½ � CD <sup>T</sup> þ 2½ � S <sup>T</sup> 

real solution of a third-degree equation [103]:

and

½ � CD <sup>3</sup> <sup>þ</sup>

1 K2, <sup>1</sup>

using a Cardin-Tartaglia formulae

where T¼ ½ � CD <sup>T</sup> þ ½ � S <sup>T</sup>. If T is kept constant in the experiments, as is common practice when Job plots are used to obtain stoichiometries, the observed displacement varies linearly with [S]<sup>T</sup> or [CD]T, but the fitting parameters are present in the form of a ratio that generates an infinite number of acceptable solutions. Consequently, it is suggested that T should be chosen in such a

Another approach lays on the assumption of a 2:1 (CD/G) complexation, in a two-step mechanism. In these circumstances, the complexation process is defined by two binding constants

Aobs <sup>¼</sup> ½ � CD <sup>f</sup> ACD <sup>þ</sup> ½ � CD � <sup>G</sup> ACD�<sup>G</sup> <sup>þ</sup> <sup>2</sup>½ � CD<sup>2</sup> � <sup>G</sup> ACD2�<sup>G</sup>

where ACD, ACD�G, and ACD2�<sup>G</sup> are the contributions of the CD and 1:1 and 2:1 CD/G complexes, with concentrations [CD], [CD � G], and [CD<sup>2</sup> � G], respectively, for the observed (experimental) physical parameter A. Using a similar procedure to that used for a 1:1 complex-

Aobs <sup>¼</sup> ACD <sup>þ</sup> ½ � CD <sup>K</sup>1ACD�<sup>G</sup> <sup>þ</sup> <sup>K</sup>1K2½ � CD ACD2�<sup>G</sup>

1 K1,1K2,<sup>1</sup>

One method for estimation of the free CD concentration is through an analytical solution of the

<sup>x</sup> <sup>¼</sup> <sup>r</sup> � <sup>1</sup> 3 <sup>a</sup> � <sup>q</sup>

� ½ � CD <sup>T</sup> K2,<sup>1</sup>

½ � <sup>G</sup> <sup>T</sup> (18)

http://dx.doi.org/10.5772/intechopen.73531

83

Interactions between Bio-Based Compounds and Cyclodextrins

�<sup>1</sup> [102].

½ � G <sup>f</sup> ¼ ½ � G <sup>T</sup> � ½ �� CD � G ½ � CD<sup>2</sup> � G (19)

½ � CD <sup>f</sup> ¼ ½ � CD <sup>T</sup> � ½ �� CD � G 2½ � CD<sup>2</sup> � G (20)

½ � CD <sup>f</sup> <sup>þ</sup> ½ �þ CD � <sup>G</sup> <sup>2</sup>½ � CD<sup>2</sup> � <sup>G</sup> (21)

<sup>1</sup> <sup>þ</sup> <sup>K</sup>1½ � CD <sup>f</sup> <sup>þ</sup> <sup>K</sup>1K2½ � CD <sup>2</sup> (22)

<sup>þ</sup> ½ � <sup>S</sup> <sup>T</sup> K2,<sup>1</sup>

f xð Þ¼ <sup>x</sup><sup>3</sup> <sup>þ</sup> ax<sup>2</sup> <sup>þ</sup> bx <sup>þ</sup> <sup>c</sup> (24)

½ �� CD ½ � CD <sup>T</sup> K1,1K2,<sup>1</sup>

<sup>r</sup> (25)

¼ 0 (23)

where f is defined as [CD � G]/[G]T.

Despite the binding process being followed by ΔA (e.g., for <sup>1</sup> H NMR, ΔA will be equal to the chemical shift of a given <sup>1</sup> H resonance), the observed ΔA for a host molecule is expressed as

$$A\_{\rm obs} = (1 - f)A\_{\rm CD,f} + fA\_{\rm CD-G} \tag{15}$$

where ACD,f and ACD�<sup>G</sup> represent the measurable physical parameter related to CD in free and complexed states, respectively.

The variation of the physical parameter in the presence and absence of a guest molecule, ΔAobs = ΔAobs � ΔACD, can be expressed as

$$
\Delta A\_{\text{obs}} = \frac{\Delta A\_{\text{CD}-G}}{[\text{CD}]\_T} \begin{bmatrix} \text{CD}-\text{G} \end{bmatrix} \tag{16}
$$

which, after some algebraic manipulation and simplification, results in [100, 101]:

$$
\Delta A\_{\rm obs} = \frac{\Delta A\_{\rm CD-G}}{2\left[\underline{\rm CD}\right]\_{\rm T}} \left\{ \left( [\rm G]\_{\rm T} + [\rm CD]\_{\rm T} + \frac{1}{K\_{\rm I}} \right) - \left( \left( [\rm G]\_{\rm T} + [\rm CD]\_{\rm T} + \frac{1}{K\_{\rm I}} \right)^{2} - 4\left( [\rm G]\_{\rm T} [\rm CD]\_{\rm T} \right) \right)^{1/2} \right\} \tag{17}
$$

It should be stressed that the application of Eq. (17) shows some drawbacks when the total concentrations of CD and guest are low and/or the binding constant is very weak, i.e., for the simplest 1:1 case, when <sup>y</sup> is sufficiently small, <sup>x</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>x</sup><sup>2</sup> � <sup>y</sup> <sup>p</sup> <sup>≈</sup> <sup>y</sup>=2x, and, consequently, Eq. (17) reduces to

Interactions between Bio-Based Compounds and Cyclodextrins http://dx.doi.org/10.5772/intechopen.73531 83

$$
\Delta A\_{\rm obs} = \frac{\Delta A\_{\rm CD-G}}{T + \left(^{1/\kappa\_1}\right)} \left[G\right]\_T \tag{18}
$$

where T¼ ½ � CD <sup>T</sup> þ ½ � S <sup>T</sup>. If T is kept constant in the experiments, as is common practice when Job plots are used to obtain stoichiometries, the observed displacement varies linearly with [S]<sup>T</sup> or [CD]T, but the fitting parameters are present in the form of a ratio that generates an infinite number of acceptable solutions. Consequently, it is suggested that T should be chosen in such a way that its value should be of the same order of magnitude than K<sup>1</sup> �<sup>1</sup> [102].

Another approach lays on the assumption of a 2:1 (CD/G) complexation, in a two-step mechanism. In these circumstances, the complexation process is defined by two binding constants K<sup>1</sup> and K2, and the corresponding mass balances are defined as

$$[\mathbf{G}]\_f = [\mathbf{G}]\_T - [\mathbf{C}D - \mathbf{G}] - [\mathbf{C}D\_2 - \mathbf{G}] \tag{19}$$

and

common method for the determination of stoichiometry is the method of continuous variation or the Job plot; the virtues and limitations of this method were recently reviewed [98], and thus

Going back to the determination of the binding constants, the most accurate way to compute K is by using the first principles. Here, for the sake of simplicity, the 1:1 and 1:2 (G/CD) stoichiometric ratios will be focused. Additionally, these examples also correspond to the large majority of complexes formed between CDs and EOs. For more complex stoichiometries, the computational treatment of the resulting equations (not shown) is not straightforward as a consequence of multicollinearity [99]. Multicollinearity causes larger standard errors in the quantities calculated and lowers statistical significance of the results. In limiting cases, several local minima may be obtained by iteration; these correspond to noticeably different combinations of the quantities calculated and may be the reason why different K values are reported for

Assuming that a 1:1 complex (CD-S) is formed, the binding constant (Eq. (2)) can be rewritten as

where ACD,f and ACD�<sup>G</sup> represent the measurable physical parameter related to CD in free and

The variation of the physical parameter in the presence and absence of a guest molecule,

½ � CD <sup>T</sup>

It should be stressed that the application of Eq. (17) shows some drawbacks when the total concentrations of CD and guest are low and/or the binding constant is very weak, i.e., for the

� ½ � G <sup>T</sup> þ ½ � CD <sup>T</sup> þ

� � !<sup>1</sup>=<sup>2</sup> <sup>8</sup>

� �<sup>2</sup>

<sup>Δ</sup>Aobs <sup>¼</sup> <sup>Δ</sup>ACD�<sup>G</sup>

which, after some algebraic manipulation and simplification, results in [100, 101]:

1 K1

ð Þ 1 � f ½ � CD <sup>T</sup> � f G½ �<sup>T</sup>

H resonance), the observed ΔA for a host molecule is expressed as

Aobs ¼ ð Þ 1 � f ACD,f þ f ACD�<sup>G</sup> (15)

� � (14)

H NMR, ΔA will be equal to the

½ � CD � G (16)

<sup>x</sup><sup>2</sup> � <sup>y</sup> <sup>p</sup> <sup>≈</sup> <sup>y</sup>=2x, and, consequently, Eq. (17)

� 4 ½ � G <sup>T</sup>½ � CD <sup>T</sup>

9 = ; (17)

1 K1

<sup>K</sup><sup>1</sup> <sup>¼</sup> <sup>f</sup>

it is not our intention to further discuss it in the present chapter.

Despite the binding process being followed by ΔA (e.g., for <sup>1</sup>

½ � G <sup>T</sup> þ ½ � CD <sup>T</sup> þ

simplest 1:1 case, when <sup>y</sup> is sufficiently small, <sup>x</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffi

� �

the same host-guest systems.

82 Cyclodextrin - A Versatile Ingredient

where f is defined as [CD � G]/[G]T.

chemical shift of a given <sup>1</sup>

complexed states, respectively.

<sup>Δ</sup>Aobs <sup>¼</sup> <sup>Δ</sup>ACD�<sup>G</sup>

reduces to

2 ½ � CD <sup>T</sup>

ΔAobs = ΔAobs � ΔACD, can be expressed as

< :

$$[\mathbf{C}D]\_f = [\mathbf{C}D]\_T - [\mathbf{C}D - \mathbf{G}] - 2[\mathbf{C}D\_2 - \mathbf{G}] \tag{20}$$

From the equilibrium constants and Eqs. (19) and (20), we can write

$$A\_{obs} = \frac{[\text{CD}]\_f A\_{\text{CD}} + [\text{CD} - \text{G}] A\_{\text{CD}-\text{G}} + 2[\text{CD}\_2 - \text{G}] A\_{\text{CD}\_2-\text{G}}}{[\text{CD}]\_f + [\text{CD} - \text{G}] + 2[\text{CD}\_2 - \text{G}]} \tag{21}$$

where ACD, ACD�G, and ACD2�<sup>G</sup> are the contributions of the CD and 1:1 and 2:1 CD/G complexes, with concentrations [CD], [CD � G], and [CD<sup>2</sup> � G], respectively, for the observed (experimental) physical parameter A. Using a similar procedure to that used for a 1:1 complexation, it is possible to write Eq. (21) as a function of [CD], that is

$$A\_{obs} = \frac{A\_{\rm CD} + [\rm CD]K\_1 A\_{\rm CD-G} + K\_1 K\_2 [\rm CD]A\_{\rm CD-G}}{1 + K\_1 [\rm CD]\_f + K\_1 K\_2 [\rm CD]^2} \tag{22}$$

On the other hand, the free CD concentration is given by

$$\left[\text{[CD]}^{3}\right] + \left(\frac{1}{K\_{2,1}} - [\text{CD}]\_{\text{T}} + 2[\text{S}]\_{\text{T}}\right) [\text{CD}]^{2} + \left(\frac{1}{K\_{1,1}K\_{2,1}} - \frac{[\text{CD}]\_{\text{T}}}{K\_{2,1}} + \frac{[\text{S}]\_{\text{T}}}{K\_{2,1}}\right) [\text{CD}] - \frac{[\text{CD}]\_{\text{T}}}{K\_{1,1}K\_{2,1}} = 0 \quad \text{(23)}$$

One method for estimation of the free CD concentration is through an analytical solution of the real solution of a third-degree equation [103]:

$$f(\mathbf{x}) = \mathbf{x}^3 + a\mathbf{x}^2 + b\mathbf{x} + c \tag{24}$$

using a Cardin-Tartaglia formulae

$$\mathbf{x} = r - \frac{1}{3}\mathbf{a} - \frac{q}{r} \tag{25}$$

where

and

$$q = \frac{1}{3}b - \frac{1}{9}a^2\tag{26}$$

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2004;76(10):1825-1845. DOI: 10.1351/pac200476101825

39(9):1033-1046. DOI: 10.1016/S0032-9592(03)00258-9

$$r = \sqrt[3]{\frac{1}{6}ab - \frac{1}{2}c - \frac{1}{27}a^3 + \sqrt{\frac{1}{27}b^3 - \frac{1}{6}abc + \frac{1}{4}c^2 + \frac{1}{27}a^3c - \frac{1}{108}a^2b^2}}\tag{27}$$
