**4.1 Determination of stoichiometry. Continuous variation method.**

Since correct reaction stoichiometry is crucial for correct binding constant determination we will study how can it be evaluated. There are different methods of calculating the

Thermodynamics as a Tool for the Optimization of Drug Binding 773

where R is the gas constant and T the absolute temperature. The free energy can be dissected

= ∆H<sup>0</sup>

On the other hand, the heat capacity (ΔCp –p subscript indicates that the system is at constant pressure-) of a reaction predicts the change of ΔH0 and ΔS0 with temperature and


T2 - ∆H<sup>0</sup> T1

T2- T1

T2 - ��� T1 ln T2 T1

In an ITC experiment a constant temperature is set, a precise amount of ligand is added to a known target molecule concentration and the heat difference is measured between reference and sample cells. To eliminate heats of mixing effects, the ligand and target as well as the reference cell contain identical buffer composition. Subsequent injections of ligand are done until no further heat of binding is observed (all sites are then bound with ligand molecules). The remaining heat generated now comes from dilution of ligand into the target solution. Data should be corrected for the heat of dilution. The heat of binding calculated for every injection is plotted versus the molar ratio of ligand to protein. Ka is related to the curve shape and binding capacity (n) determined from the ratio of ligand to target at the equivalence point of the curve. Data must be fitted to a binding model. The type of binding must be known from other experimental techniques. Here, we will study the simplest model with a single site. Equations 6 and 7 can be rearranged to find the following relation

Total ligand concentration is known and can be represented as (remember that we are

[L]total= [L] + υ[P]total (34)

<sup>+</sup> <sup>1</sup>� <sup>υ</sup><sup>+</sup> [L]total [P]total

> + <sup>1</sup> Ka[P]total

+ 1� 2 - 4 [L]total

[L]total [P]total

V = υ[P]total∆H<sup>0</sup>

+ <sup>1</sup> Ka[P]total

+ 1� -��

(30)

(31)

(32)

(1-υ)[L] (33)

= 0 (35)

V (37)

[P]total � (36)

∆G <sup>0</sup>

∆Cp= <sup>∆</sup>H<sup>0</sup>

<sup>∆</sup>Cp= ���

into enthalpic and entropic components by:

can be expressed as:

between υ and Ka:

assuming m=n=1):

Solving for υ:

Ka= <sup>υ</sup>

υ2 - � [L]total [P]total

> + <sup>1</sup> Ka[P]total

The total heat content (Q) in the sample cell at volume (V) can be defined as:

Q = [PL]∆H<sup>0</sup>

Combining equations 33 and 34 gives:

υ = <sup>1</sup>

<sup>2</sup> ��[L]total [P]total

or

stoichoimetry: continuous variation method, slope ratio method, mole ratio method, being the first one, the continuous variation method the most popular. In order to determine the stoichiometry by this method the concentration of the produced complex (or any property proportional to it) is plotted versus the mole fraction ligand ([L]total/([P]total+[L]total)) over a number of tritation steps where the sum of [P]total and [L]total is kept constant (α) changing [L]total from 0 to α. The maxima of this plot (known as Job's plot, (Job 1928; Ingham 1975)) indicates the stoichiometry of the binding reaction: 1:1 is indicated by a maximum at 0.5 since this value corresponds to n/(n+m). For the understanding of the theoretical background of the method, it is important to remember equations 2 and 5; notice that:

$$\left[\mathbf{P}\right]\_{\text{total}} = \left[\mathbf{P}\right] + \text{m}\left[\mathbf{P}\_{\text{m}}\mathbf{L}\_{\text{m}}\right] \tag{20}$$

$$\left[\mathbf{L}\right]\_{\text{total}} = \left[\mathbf{L}\right] + \mathbf{n} \left[\mathbf{P}\_{\text{m}} \mathbf{L}\_{\text{m}}\right] \tag{21}$$

$$\mathbf{u} = \begin{bmatrix} \mathbf{L} \end{bmatrix}\_{\text{total}} + \begin{bmatrix} \mathbf{P} \end{bmatrix}\_{\text{total}} \tag{22}$$

$$\mathbf{x}\_{\text{X}} = \frac{\begin{bmatrix} \mathbf{L} \end{bmatrix}\_{\text{total}}}{\begin{bmatrix} \mathbf{P} \end{bmatrix}\_{\text{total}} + \begin{bmatrix} \mathbf{L} \end{bmatrix}\_{\text{total}}} \tag{23}$$

$$\mathbf{y} = [\mathbf{P}\_{\mathrm{m}} \mathbf{L}\_{\mathrm{m}}] \tag{24}$$

Substitution of [P]total and [L]total by the functions of α and x from equation 23 and 24 yields:

$$\left[\mathbf{P}\right]\_{\text{total}} = \mathbf{a} \cdot \mathbf{c} \mathbf{x} \tag{25}$$

$$\left[\mathbf{L}\right]\_{\text{total}} = \mathbf{c}\mathbf{x} \tag{26}$$

from equations 2, 5, 20, 21, 24, 25, 26:

$$\mathbf{y} = \mathbf{K}\_{\mathbf{a}} (\mathbf{a} \cdot \mathbf{m} \mathbf{y} \ -\alpha \mathbf{x})^{\mathbf{m}} (\alpha \mathbf{x} \ -\mathbf{n} \mathbf{y})^{\mathbf{n}} \tag{27}$$

Equation 27 is differentiated, and the dy/dx substituted by zero to obtain the x-coordinate at the maximum:

$$\chi = \frac{\mathfrak{n}}{\mathfrak{n} + \mathfrak{m}} \tag{28}$$

This equation shows the correlation between stoichiometry and the x-coordinate at the maximum in Job's plot. That's why a maximum at x = 0.5 means a 1:1 stoichiometry (n = m = 1). In the case of 1:2 the maximum would be at x = 1/3.

#### **4.2 Calorimetry**

Isothermal titration calorimetry (ITC) is a useful tool for the characterization of thermodynamics and kinetics of ligands binding to macromolecules. With this method the rate of heat flow induced by the change in the composition of the target solution by tritation of a ligand (or vice versa) is measured. This heat is proportional to the total amount of binding. Since the technique measures heat directly, it allows simultaneous determination of the stoichiometry (n), the binding constant (Ka) and the enthalpy (ΔH0) of binding. The free energy (ΔG0) and the entropy (ΔS0) are easily calculated from ΔH0 and Ka. Note that the binding constant is related to the free energy by:

$$
\Delta \mathbf{G}^{\;0} = \text{-RT} \ln \mathbf{K}\_{\mathbf{a}} \tag{29}
$$

where R is the gas constant and T the absolute temperature. The free energy can be dissected into enthalpic and entropic components by:

$$
\Delta \mathbf{G}^{\;0} = \Delta \mathbf{H}^{0} \text{-} \mathbf{T} \Delta \mathbf{S}^{0} \tag{30}
$$

On the other hand, the heat capacity (ΔCp –p subscript indicates that the system is at constant pressure-) of a reaction predicts the change of ΔH0 and ΔS0 with temperature and can be expressed as:

$$
\Delta \mathbf{C\_p} = \frac{\Delta \mathbf{H^0\_{T2} \cdot \Delta \mathbf{H^0\_{T1}}}}{T\_2 \cdot T\_1} \tag{31}
$$

or

772 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

stoichoimetry: continuous variation method, slope ratio method, mole ratio method, being the first one, the continuous variation method the most popular. In order to determine the stoichiometry by this method the concentration of the produced complex (or any property proportional to it) is plotted versus the mole fraction ligand ([L]total/([P]total+[L]total)) over a number of tritation steps where the sum of [P]total and [L]total is kept constant (α) changing [L]total from 0 to α. The maxima of this plot (known as Job's plot, (Job 1928; Ingham 1975)) indicates the stoichiometry of the binding reaction: 1:1 is indicated by a maximum at 0.5 since this value corresponds to n/(n+m). For the understanding of the theoretical background of the method, it is important to remember equations 2 and 5; notice that:

[P]total= [P] + m[PmLn] (20)

[L]total= [L] + n[PmLn] (21)

[L]total [P]total + [L]total

Substitution of [P]total and [L]total by the functions of α and x from equation 23 and 24 yields:

[P]total= α - αx (25)

[L]total= αx (26)

Equation 27 is differentiated, and the dy/dx substituted by zero to obtain the x-coordinate at

x = <sup>n</sup>

This equation shows the correlation between stoichiometry and the x-coordinate at the maximum in Job's plot. That's why a maximum at x = 0.5 means a 1:1 stoichiometry (n = m

Isothermal titration calorimetry (ITC) is a useful tool for the characterization of thermodynamics and kinetics of ligands binding to macromolecules. With this method the rate of heat flow induced by the change in the composition of the target solution by tritation of a ligand (or vice versa) is measured. This heat is proportional to the total amount of binding. Since the technique measures heat directly, it allows simultaneous determination of the stoichiometry (n), the binding constant (Ka) and the enthalpy (ΔH0) of binding. The free energy (ΔG0) and the entropy (ΔS0) are easily calculated from ΔH0 and Ka. Note that the

∆G <sup>0</sup>

(αx - ny)<sup>n</sup>

x =

y = Ka(α - my - αx)<sup>m</sup>

from equations 2, 5, 20, 21, 24, 25, 26:

= 1). In the case of 1:2 the maximum would be at x = 1/3.

binding constant is related to the free energy by:

the maximum:

**4.2 Calorimetry** 

α = [L]total + [P]total (22)

y = [PmLn] (24)

(23)

(27)

<sup>n</sup> <sup>+</sup> <sup>m</sup> (28)

= -RT ln Ka (29)

$$
\Delta \mathbf{C\_{p}} = \frac{\Delta \mathbf{S^{0}\_{T2}} \cdot \Delta \mathbf{S^{0}\_{T1}}}{\ln \frac{\mathbf{r\_{2}}}{\mathbf{r\_{1}}}} \tag{32}
$$

In an ITC experiment a constant temperature is set, a precise amount of ligand is added to a known target molecule concentration and the heat difference is measured between reference and sample cells. To eliminate heats of mixing effects, the ligand and target as well as the reference cell contain identical buffer composition. Subsequent injections of ligand are done until no further heat of binding is observed (all sites are then bound with ligand molecules). The remaining heat generated now comes from dilution of ligand into the target solution. Data should be corrected for the heat of dilution. The heat of binding calculated for every injection is plotted versus the molar ratio of ligand to protein. Ka is related to the curve shape and binding capacity (n) determined from the ratio of ligand to target at the equivalence point of the curve. Data must be fitted to a binding model. The type of binding must be known from other experimental techniques. Here, we will study the simplest model with a single site. Equations 6 and 7 can be rearranged to find the following relation between υ and Ka:

$$\mathbf{K}\_{\mathbf{a}} = \frac{\mathbf{o}}{\text{(1-o)[L]}} \tag{33}$$

Total ligand concentration is known and can be represented as (remember that we are assuming m=n=1):

$$\left[\mathbf{L}\right]\_{\text{total}} = \left[\mathbf{L}\right] + \mathbf{u}\left[\mathbf{P}\right]\_{\text{total}}\tag{34}$$

Combining equations 33 and 34 gives:

$$\mathbf{U}^{\frac{1}{2}} - \left(\frac{[\mathbf{I}\mathbf{J}]\_{\mathrm{total}}}{[\mathbf{I}\mathbf{P}]\_{\mathrm{total}}} + \frac{1}{\mathbf{K}\_{\mathrm{a}}[\mathbf{P}]\_{\mathrm{total}}} + \mathbf{1}\right) \mathbf{U} + \frac{[\mathbf{I}\mathbf{J}]\_{\mathrm{total}}}{[\mathbf{P}]\_{\mathrm{total}}} = \mathbf{0} \tag{35}$$

Solving for υ:

$$\mathbf{U} = \frac{1}{2} \left\lfloor \left( \frac{\text{[L]}\_{\text{total}}}{\text{[P]}\_{\text{total}}} + \frac{1}{\text{K}\_{\text{l}} \text{[P]}\_{\text{total}}} + 1 \right) \cdot \sqrt{\left( \frac{\text{[L]}\_{\text{total}}}{\text{[P]}\_{\text{total}}} + \frac{1}{\text{K}\_{\text{l}} \text{[P]}\_{\text{total}}} + 1 \right)^{2} \cdot \frac{4 \text{ [L]}\_{\text{total}}}{\text{[P]}\_{\text{total}}}} \right\rfloor \tag{36}$$

The total heat content (Q) in the sample cell at volume (V) can be defined as:

$$\mathbf{Q} = \text{[PL]}\boldsymbol{\Delta}\mathbf{H}^0 \mathbf{V} = \mathbf{v}[\mathbf{P}]\_{\text{total}} \boldsymbol{\Delta}\mathbf{H}^0 \mathbf{V} \tag{37}$$

Thermodynamics as a Tool for the Optimization of Drug Binding 775

which is the direct plot expressed in terms of spectrophotometric observation. Note that the

The free ligand concentration is actually unknown. The known concentrations are [P]total to which a known [L]total is added. In a similar way as shown above for [P]total, [L]total can be

From equations 44 and 45 a complete description of the system is obtained. If [L]total >>[P]total we will have that [L]total ≈ [L] from equation 45, equation 44 can be then analysed with this approximation. With this first rough estimate of Ka, equation 45 can be solved for the [L] value for each [L]total. These values can be used in equation 44 to obtain an improved estimation of Ka, and this process should be repeated until the solution for Ka reaches a constant value. Equation 44 can be solved graphically using any of the plots presented in

Fluorescence spectroscopy is a widely used tool in biochemistry due to its ease, sensitivity to local environmental changes and ability to describe target-ligand interactions qualitatively and quantitatively in equilibrium conditions. In this technique the fluorophore molecule senses changes in its local environment. To analyse ligand-target interactions it is possible to take advantage of the nature of ligands, excepcionally we can find molecules which are essentially non or weakely fluorescent in solution but show intense fluorescence upon binding to their targets (that is the case, for example, of colchicines and some of its analogues). Fluorescence moieties such as fluorescein can be also attached to naturally non-fluorescent ligands to make used of these methods. The fluorescent dye may influence the binding, so an essential control with any tagged molecule is a competition experiment with the untagged molecule. Finally, in a few favourable cases the intrinsic tryptophan fluorescence of a protein changes when a ligand binds, usually decreasing (fluorescence quenching). Again, increasing concentrations of ligand to a fixed concentration of target (or vice versa) are incubated at controlled temperature and fluorescence changes measured until saturation is reached. Binding constant can be determined by fitting data according to equation 11 (Scatchard plot). From fluorescence data

Fmax

If free ligand has an appreciable fluorescence as compared to ligand bound to its target, then the fluorescence enhancement factor (Q) should be determined. Q is defined as (Mas &

> Q = Fbound Ffree

To determine it, a reverse titration should be done. The enhancement factor can be obtained from the intercept of linear plot of 1/((F/F0)-1) against 1/P, where F and F0 are the observed fluorescence in the presence and absence of target, respectively. Once it is known, the concentration of complex can be determine from a fluorescence titration experiment using:

1 + Ka [L] (45)

(46)


dependence of ΔAbs/l on [L] is the same as the one shown in equation 7.

[L]total= [L] [P]total Ka [L]

written as:

section 3.1.

**4.3.2 Fluorescence** 

Colman 1985):

(F), υ can be calculated from the relantionship:

υ = Fmax- F

where ΔH0 is the heat of binding of the ligand to its target. Substituing equation 36 into 37 yields:

$$\mathbf{Q} = \frac{\left[\text{P}\right]\_{\text{total}}\text{\&P}^{0}\text{V}}{2} \left| \left( \frac{\text{[L}\_{\text{total}}}{\text{[P]}\_{\text{total}}} + \frac{1}{\text{K}\_{\text{a}}\text{[P]}\_{\text{total}}} + 1 \right) - \sqrt{\left( \frac{\text{[L}\_{\text{total}}}{\text{[P]}\_{\text{total}}} + \frac{1}{\text{K}\_{\text{a}}\text{[P]}\_{\text{total}}} + 1 \right)^{2} - \frac{4 \left[\text{L}\right]\_{\text{total}}}{\left[\text{P}\right]\_{\text{total}}}} \right| \tag{38}$$

Therefore Q is a function of Ka and ΔH0 (and n, but here we considered it as 1 for simplicity) since [P]total, [L]total and V are known for each experiment.

### **4.3 Optical spectroscopy**

The goal to be able to determine binding affinity is to measure the equilibrium concentration of the species implied over a range of concentrations of one of the reactants (P or L). Measuring one of them should be sufficient as total concentrations are known and therefore the others can be calculated by difference from total concentrations and measured equilibrium concentration of one of the species. Plotting the concentration of the complex (PL) against the free concentration of the varying reactant, the binding constant could be calculated.

#### **4.3.1 Absorbance**

As an example a 1:1 stoichiometry model will be shown, wherein the Lambert-Beer law is obeyed by all the reactants implied. To use this technique we should ensured that the complex (PL) has a significantly different absorption spectrum than the target molecule (P) and a wavelenght at which both molar extinction coefficients are different should be selected. At these conditions the absorbance of the target molecule in the absence of ligand will be:

$$\mathbf{Abs}\_0 = \mathbf{c}\_\mathbf{P} \mathbf{l} \begin{bmatrix} \mathbf{P} \end{bmatrix}\_{\text{total}} \tag{39}$$

If ligand is added to a fixed total target concentration, the absorbance of the mix can be written as:

$$\text{Abs}\_{\text{mix}} = \varepsilon\_{\text{P}} \mathbf{l} \left[ \mathbf{P} \right] + \varepsilon\_{\text{L}} \mathbf{l} \left[ \mathbf{L} \right] + \varepsilon\_{\text{PL}} \mathbf{l} \left[ \mathbf{P} \mathbf{L} \right] \tag{40}$$

Since [P]total = [P] + [PL] and [L]total = [L] + [PL], equation 40 can be rewritten as:

$$\text{Abs}\_{\text{mix}} = \varepsilon\_{\text{P}} \text{ l [P]}\_{\text{total}} + \varepsilon\_{\text{L}} \text{ l [L]}\_{\text{total}} + \Delta \varepsilon \text{ l [PL]} \tag{41}$$

where Δε = εPL-εP-εL. If the blank solution against which samples are measured contains [L]total, then the observed absorbance would be:

$$\mathbf{Abs}\_{\rm obs} = \varepsilon\_{\rm P} \mathbf{l} \left[ \mathbf{P} \right]\_{\rm total} + \Delta \varepsilon \mathbf{l} \left[ \mathbf{PL} \right] \tag{42}$$

Substracting equation 39 from 42 and incorporating Ka (equation 5):

$$
\Delta \text{Abs} = \mathsf{K}\_{\text{a}} \,\, \Delta \varepsilon \,\mathrm{l} \,\, \left[\mathrm{P}\right] \left[\mathrm{L}\right] \tag{43}
$$

[P]total can be written as [P]total = [P](1+Ka[L]) which included in equation 43 yields:

$$\frac{\Delta \text{Abs}}{\text{l}} = \frac{[\text{P}]\_{\text{total}} \text{ K}\_{\text{a}} \Delta e \text{ [L]}}{\text{1} + [\text{K}\_{\text{a}} \text{ [L]}]} \tag{44}$$

which is the direct plot expressed in terms of spectrophotometric observation. Note that the dependence of ΔAbs/l on [L] is the same as the one shown in equation 7.

The free ligand concentration is actually unknown. The known concentrations are [P]total to which a known [L]total is added. In a similar way as shown above for [P]total, [L]total can be written as:

$$\begin{bmatrix} \mathbf{L} \end{bmatrix}\_{\text{total}} = \begin{bmatrix} \mathbf{L} \end{bmatrix} \begin{array}{c} \begin{bmatrix} \mathbf{F} \end{bmatrix}\_{\text{total}} \begin{array}{c} \mathbf{K}\_{\text{a}} \begin{bmatrix} \mathbf{L} \end{bmatrix} \end{array} \tag{45}$$

From equations 44 and 45 a complete description of the system is obtained. If [L]total >>[P]total we will have that [L]total ≈ [L] from equation 45, equation 44 can be then analysed with this approximation. With this first rough estimate of Ka, equation 45 can be solved for the [L] value for each [L]total. These values can be used in equation 44 to obtain an improved estimation of Ka, and this process should be repeated until the solution for Ka reaches a constant value. Equation 44 can be solved graphically using any of the plots presented in section 3.1.

#### **4.3.2 Fluorescence**

774 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

where ΔH0 is the heat of binding of the ligand to its target. Substituing equation 36 into 37

+ 1� - ��

Therefore Q is a function of Ka and ΔH0 (and n, but here we considered it as 1 for simplicity)

The goal to be able to determine binding affinity is to measure the equilibrium concentration of the species implied over a range of concentrations of one of the reactants (P or L). Measuring one of them should be sufficient as total concentrations are known and therefore the others can be calculated by difference from total concentrations and measured equilibrium concentration of one of the species. Plotting the concentration of the complex (PL) against the free concentration of the varying reactant, the binding constant could be

As an example a 1:1 stoichiometry model will be shown, wherein the Lambert-Beer law is obeyed by all the reactants implied. To use this technique we should ensured that the complex (PL) has a significantly different absorption spectrum than the target molecule (P) and a wavelenght at which both molar extinction coefficients are different should be selected. At these conditions the absorbance of the target molecule in the absence of ligand

 Abs0= ε<sup>P</sup> l [P]total (39) If ligand is added to a fixed total target concentration, the absorbance of the mix can be

Absmix= εP l [P] + εL l [L] + εPL l [PL] (40)

 Absmix= εP l [P]total+ εL l [L]total+ ∆ε l [PL] (41) where Δε = εPL-εP-εL. If the blank solution against which samples are measured contains

Absobs= ε<sup>P</sup> l [P]total+ Δε l [PL] (42)

<sup>l</sup> <sup>=</sup> [P]total Ka ∆ε [L]

∆Abs = Ka ∆ε l [P] [L] (43)

� � Ka [L] (44)

Since [P]total = [P] + [PL] and [L]total = [L] + [PL], equation 40 can be rewritten as:

[P]total can be written as [P]total = [P](1+Ka[L]) which included in equation 43 yields:

ΔAbs

Substracting equation 39 from 42 and incorporating Ka (equation 5):

[L]total, then the observed absorbance would be:

[L]total [P]total

+ <sup>1</sup> Ka[P]total + 1� 2

 - 4 [L]total [P]total

� (38)

yields:

calculated.

will be:

written as:

**4.3.1 Absorbance** 

Q = [P]total∆H<sup>0</sup>

**4.3 Optical spectroscopy** 

V <sup>2</sup> ��[L]total [P]total

since [P]total, [L]total and V are known for each experiment.

+ <sup>1</sup> Ka[P]total

> Fluorescence spectroscopy is a widely used tool in biochemistry due to its ease, sensitivity to local environmental changes and ability to describe target-ligand interactions qualitatively and quantitatively in equilibrium conditions. In this technique the fluorophore molecule senses changes in its local environment. To analyse ligand-target interactions it is possible to take advantage of the nature of ligands, excepcionally we can find molecules which are essentially non or weakely fluorescent in solution but show intense fluorescence upon binding to their targets (that is the case, for example, of colchicines and some of its analogues). Fluorescence moieties such as fluorescein can be also attached to naturally non-fluorescent ligands to make used of these methods. The fluorescent dye may influence the binding, so an essential control with any tagged molecule is a competition experiment with the untagged molecule. Finally, in a few favourable cases the intrinsic tryptophan fluorescence of a protein changes when a ligand binds, usually decreasing (fluorescence quenching). Again, increasing concentrations of ligand to a fixed concentration of target (or vice versa) are incubated at controlled temperature and fluorescence changes measured until saturation is reached. Binding constant can be determined by fitting data according to equation 11 (Scatchard plot). From fluorescence data (F), υ can be calculated from the relantionship:

$$\mathbf{u} = \frac{\mathbf{F}\_{\text{max}} \mathbf{\ast} \mathbf{F}}{\mathbf{F}\_{\text{max}}} \tag{46}$$

If free ligand has an appreciable fluorescence as compared to ligand bound to its target, then the fluorescence enhancement factor (Q) should be determined. Q is defined as (Mas & Colman 1985):

$$\mathbf{Q} = \frac{\text{F}\_{\text{bound}}}{\text{F}\_{\text{fore}}} \text{-1} \tag{47}$$

To determine it, a reverse titration should be done. The enhancement factor can be obtained from the intercept of linear plot of 1/((F/F0)-1) against 1/P, where F and F0 are the observed fluorescence in the presence and absence of target, respectively. Once it is known, the concentration of complex can be determine from a fluorescence titration experiment using:

Thermodynamics as a Tool for the Optimization of Drug Binding 777

Microtubule stabilizing agents (MSA) comprise a class of drugs that bind to microtubules and stabilize them against disassembly. During the last years, several of these compounds have been approved as anticancer agents or submitted to clinical trials. That is the case of taxanes (paclitaxel, docetaxel) or epothilones (ixabepilone) as well as discodermolide (reviewed in (Zhao *et al.* 2009)). Nevertheless, anticancer chemotherapy has still unsatisfactory clinical results, being one of the major reasons for it the development of drug resistance in treated patients (Kavallaris 2010). Thus one interesting issue in this field is drug optimization with the aim of improving the potential for their use in clinics: minimizing

Our group has studied the influence of different chemical modifications on taxane and epothilone scaffolds in their binding affinities and the consequently modifications in ligand properties like citotoxicity. The results from these studies firmly suggest thermodynamic

Epothilones are one of the most promising natural products discovered with paclitaxel-like activity. Their advantages come from the fact that they can be produced in large amounts by fermentation (epothilones are secondary metabolites from the myxobacterium *Sorangiun celulosum*), their higher solubility in water, their simplicity in molecular architecture which makes possible their total synthesis and production of many analogs, and their effectiveness

The structure affinity-relationship of a group of chemically modified epothilones was studied. Epothilones derivatives with several modifications in positions C12 and C13 and

Epothilone binding affinities to microtubules were measured by displacement of Flutax-2, a fluorescent taxoid probe (fluorescein tagged paclitaxel). Both epothilones A and B binding constants were determined by direct sedimentation which further validates Flutax-2

All compounds studied are related by a series of single group modifications. The measurement of the binding affinity of such a series can be a good approximation of the incremental binding energy provided by each group. Binding free energies are easily calculated from binding constants applying equation 29. The incremental free energies (ΔG0)

change associated with the modification of ligand L into ligand S is defined as:

against multi-drug resistant cells due to they are worse substrates for P-glycoprotein.

side-effects, overcoming resistances or enhancing their potency.

parameters as key clues for drug optimization.

the side chain in C15 were used in this work.

Fig. 1. Epothilone atom numbering.

displacement method.

**5. Drug optimization** 

**5.1 Epothilones** 

$$\begin{bmatrix} \text{PL} \end{bmatrix} = \begin{bmatrix} \text{L} \end{bmatrix}\_{\text{total}} \frac{\text{(F/F\_0)} \cdot 1}{\text{Q} \cdot 1} \tag{48}$$

Thus the binding constant can be determined from the Scatchard plot as described above.

### **4.3.3 Fluorescence anisotropy**

Fluorescence anisotropy measures the rotational diffusion of a molecule. The effective size of a ligand bound to its target usually increases enormously, thus restricting its motion considerably. Changes in anisotropy are proportional to the fraction of ligand bound to its target. Using suitable polarizers at both sides of the sample cuvette, this property can be measured. In a tritation experiment similar to the ones described above, the fraction of ligand bound (XL=[PL]/[L]total) is determined from:

$$\mathbf{X}\_{\rm L} = \frac{\mathbf{r} \cdot \mathbf{r}\_0}{\mathbf{r}\_{\rm max} \cdot \mathbf{r}\_0} \tag{49}$$

where r is the anisotropy of ligand in the presence of the target molecule, r0 is the anisotropy of ligand in the absence of target and rmax is the anisotropy of ligand fully bound to its target (note that equation 49 can be used only in the case where ligand fluorescence intensity does not change, otherwise appropriate corrections should be done, see (Lakowicz 1999)). [P] can be calculated from:

$$\mathbf{I}\begin{bmatrix}\mathbf{P}\end{bmatrix} = \begin{bmatrix}\mathbf{P}\end{bmatrix}\_{\text{total}} \mathbf{-}\begin{aligned} \mathbf{X}\_{\text{L}}\begin{bmatrix}\mathbf{L}\end{bmatrix}\_{\text{total}} \end{aligned} \tag{50}$$

The binding constant can be determined from the hyperbola:

$$\mathcal{K}\_{\rm L} = \frac{\mathcal{K}\_{\rm u} \text{[P]}}{1 + \mathcal{K}\_{\rm u} \text{[P]}} \tag{51}$$

#### **4.4 Competition methods**

The characterization of a ligand binding let us determine the binding constant of any other ligand competing for the same binding site. Measurements of ligand (L), target (P), reference ligand (R) and both complexes (PR and PL) concentrations in the equilibrium permit the calculation of the binding constant (KL) from equation 53 (see below) as the binding constant of the reference ligand (KR) is already known.

$$\text{L} + \text{R} + \text{P} \leftrightarrow \text{PL} + \text{PR} \tag{52}$$

$$\mathbf{K}\_{\rm L} = \mathbf{K}\_{\rm R} \frac{[\rm PL][\rm R]}{[\rm L][\rm PR]} \tag{53}$$

In the case that the reference ligand has been characterized due to the change of a ligand physical property (i.e. fluorescence, absorbance, anisotropy) upon binding, would permit us also following the displacement of this reference ligand from its site by competition with a ligand "blind" to this signal (Diaz & Buey 2007). In this kind of experiment equimolar concentrations of the reference ligand and the target molecule are incubated, increasing concentrations of the problem ligand added and the appropiate signal measured. It is possible then to determine the concentration of ligand at which half the reference ligand is bound to its site (EC50). Thus KL is calculated from:

$$\mathbf{K}\_{\rm L} = \frac{1 + [\rm R]\mathbf{K}\_{\rm R}}{\rm EC\_{50}} \tag{54}$$
