**2.1.3 Hydrogen bonds**

A hydrogen bond is an interaction between a proton donor group (a hydrogen atom covalently bound to an electronegative atom -e.g. F, O, N, S-) and a proton acceptor atom (another electronegative atom). It is a very important interaction responsible for the structure and properties of water, as well as the structure and properties of biological macromolecules (e.g. hydrogen bonds are responsible of specific base-pair formation in the DNA double helix).

Hydrogen bonds are fundamentally electrostatic interactions. The relatively electronegative atom to which the hydrogen atom is covalently bonded pulls electron density away from the hydrogen atom so that it develops a partial positive charge (δ+). Thus, it can interact with an atom having a partial negative charge (δ- ) through an electrostatic interaction. However, this interaction is more than just an ionic or dipole-dipole interaction between the donor and the acceptor groups. Here, the distance between the hydrogen and acceptor atoms is less than the sum of their respective van der Waals radii.

Hydrogen bonds are directional toward the electronegative atom. The strongest hydrogen bonds have a tendency to be approximately straight, such that the proton donor group, the hydrogen atom, and the acceptor atom lie along a straight line, with significant weakening of the interaction if they are not colinear. They are somewhat longer than are covalent bonds. Hydrogen bonds are constantly being made and remade. Their half-life is about 10 seconds. These bonds have only 5% or so of the strength of covalent bonds. They have energies of 5-15 kJ/mol compared with approximately 420 kJ/mol for a carbon-hydrogen covalent bond. However, when many hydrogen bonds can form between two molecules (or parts of the same molecule), the resulting union can be sufficiently strong as to be quite stable. Examples of multiple hydrogen bonds are widely found in biological systems, they hold secondary structures of polypeptides, help in binding of enzymes to their substrate or antibodies to their antigen, help also transcription factors bind to each other or to DNA.

### **2.1.4 Hydrophobic interactions**

Hydrophobic interactions result when non-polar molecules are in a polar solvent (e.g. water). The non-polar molecules group together to exclude water so that they minimize the

Thermodynamics as a Tool for the Optimization of Drug Binding 769

Determining υ is often easy in spectrophotometric manipulation as will be discussed later. Given υ, equation 5 can then be solved for [PL] and the answer substituted into equation 6

υ = Ka[L]

This equation is the 1:1 binding isotherm also known as the Langmuir isotherm or the "direct" plot. Its functional form is a rectangular hyperbola whose midpoint will yield Ka. Chemical interpretation of 1:1 binding is that the target P has a single "binding site", as has the ligand L; and when the complex PL forms, no further sites are available for the binding of any additional ligand. To test the 1:1 stoichoimetry equation 7 may be rearrange into a linear plotting form. Since υ is the bound fraction, then 1-υ is the free one: (1- υ) =

This log-log plot should be linear with a slope of one if the stoichoimetry is really 1:1. This is called a Hill plot. Equation 7 can be also rearranged to three different non-logarithmic linear plotting forms. Taking simply the reciprocal of the equation yields the double-reciprocal

In spectroscopic studies this plot is commonly known as the Benesi-Hildebrand plot (Benesi

<sup>υ</sup> = [L] + <sup>1</sup>

And the third plotting of υ/[L] agains υ, sometimes called Scatchard plot (Scatchard 1949):

Linearity in all of these plots is a necessary condition if the 1:1 model is valid; and from the parameters of equations Ka can be evaluated. Usually υ is not measured directly but rather some experimental quantity related to it, so that the interpretation of the plots depends on

Most biological systems tend to have more than one binding site, that is the case of many systems of small molecules binding to proteins. In these cases we may consider that n ligands may bind to a single target molecule. The average number of ligand molecules

> b = Ltotal – [L] Ptotal

Ka

1 <sup>υ</sup> <sup>=</sup> <sup>1</sup>

[L]

υ

Another plot is that of [L]/υ against [L] which is expected to be linear:

= [PL]

[PL] + [P] (6)

<sup>1</sup> <sup>+</sup> Ka[L] (7)

1- <sup>υ</sup> <sup>=</sup>log [L] + log Ka (8)

Ka[L] + 1 (9)

[L] <sup>=</sup>υKa + Ka (11)

(10)

(12)

υ = Lbound Ptotal

to obtain the quantitative 1:1 stoichiometric model:

1/(1+Ka[L]). Thus υ/(1- υ) = Ka[L], and:

log <sup>υ</sup>

plot (used by plotting 1/υ against 1/[L]):

the particular experimental methodology.

bound per target molecule (b) is defined as:

& Hildebrand 1949).

**3.2 Multi-site binding** 

surface area in contact with the polar solvent. Unlike the non-covalent interactions mentioned above, which are pairwise interactions between atoms or parts of molecules, the nature of the hydrophobic interaction is very different. It involves a considerable number of (water) molecules, and does not arise from a direct force between the non-polar molecules.

Nonpolar molecules are not good acceptors of hydrogen bonds. When a non-polar molecule is placed in water, the hydrogen bonding network of water is disrupted. Water molecules must reorganize around the solute and make a kind of cage, similar to the structure of water in ice, in order to gain back the broken hydrogen bonds. This reorganization results in a considerable loss in the configurational entropy of water and therefore, in an increase in the free energy. If there are more than one such non-polar molecules, the configuration in which they are clustered together is preferred because now the hydrogen bonding network of water is disrupted in just one (albeit bigger) pocket, rather than in several small pockets. Therefore, the entropy of water is larger when the non-polar molecules are clustered together, leading to a decrease in the free energy.

Hydrophobic interactions have strengths comparable in energy to hydrogen bonds.

### **3. Binding constants**

Most drugs have a non-covalent binding to their targets, thus these interactions are of great importance for our studies. Measurements of equilibrium constants, their dependence with temperature, the determination of stoichiometry, provide main information on the mechanism of the chemical process involved. The basic process can be taken out of the association of ligand (or ligands) to its target. The binding reaction can be writen as follows:

$$\text{mP} + \text{nL} \leftrightarrow \text{P}\_{\text{m}}\text{L}\_{\text{n}}\tag{2}$$

Regardless of mechanism, every reversible reaction reaches an equilibrium distribution of reactants and products. At some point the rates of the opposing reactions (association and dissociation in our case) become equal and there would no longer be any change in the concentration of the molecules implied.

$$\mathbf{v}\_{\rm ass} = \mathbf{k}\_{\rm ass} \mathbf{[P]}^{\rm m} [\mathbf{L}]^{\rm n} \tag{3}$$

$$\mathbf{v}\_{\rm diss} = \mathbf{k}\_{\rm diss} \mathbf{[P}\_{\rm m} \mathbf{I}\_{\rm m}] \tag{4}$$

Under these conditions (vass = vdiss):

$$\frac{\mathbf{k\_{ass}}}{\mathbf{k\_{diss}}} = \frac{[\mathbf{P\_{m}}\mathbf{I\_{n}}]}{[\mathbf{P}]^{m}[\mathbf{L}]^{n}} = \mathbf{K\_{a}}\tag{5}$$

that will be the equilibrium association constant assuming that activities are equal to concentrations.

In this section we will discuss the cases for one single site in the target, multiple sites with same affinities and multiple sites with different affinities.

#### **3.1 One-site binding**

In the simplest case, where there is only one site per target molecule, n and m are 1. It is possible to define the fraction of occupied binding sites (υ) as:

Thermodynamics as a Tool for the Optimization of Drug Binding 769

768 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

surface area in contact with the polar solvent. Unlike the non-covalent interactions mentioned above, which are pairwise interactions between atoms or parts of molecules, the nature of the hydrophobic interaction is very different. It involves a considerable number of (water) molecules, and does not arise from a direct force between the non-polar molecules. Nonpolar molecules are not good acceptors of hydrogen bonds. When a non-polar molecule is placed in water, the hydrogen bonding network of water is disrupted. Water molecules must reorganize around the solute and make a kind of cage, similar to the structure of water in ice, in order to gain back the broken hydrogen bonds. This reorganization results in a considerable loss in the configurational entropy of water and therefore, in an increase in the free energy. If there are more than one such non-polar molecules, the configuration in which they are clustered together is preferred because now the hydrogen bonding network of water is disrupted in just one (albeit bigger) pocket, rather than in several small pockets. Therefore, the entropy of water is larger when the non-polar molecules are clustered

Hydrophobic interactions have strengths comparable in energy to hydrogen bonds.

Most drugs have a non-covalent binding to their targets, thus these interactions are of great importance for our studies. Measurements of equilibrium constants, their dependence with temperature, the determination of stoichiometry, provide main information on the mechanism of the chemical process involved. The basic process can be taken out of the association of ligand (or ligands) to its target. The binding reaction can be

Regardless of mechanism, every reversible reaction reaches an equilibrium distribution of reactants and products. At some point the rates of the opposing reactions (association and dissociation in our case) become equal and there would no longer be any change in the

vdiss= kdiss[PmLn] (4)

= [PmLn]

that will be the equilibrium association constant assuming that activities are equal to

In this section we will discuss the cases for one single site in the target, multiple sites with

In the simplest case, where there is only one site per target molecule, n and m are 1. It is

kass kdiss

mP + nL ↔ PmLn (2)

(3)

[P]m[L]n = Ka (5)

together, leading to a decrease in the free energy.

concentration of the molecules implied.

Under these conditions (vass = vdiss):

vass <sup>=</sup> kass[P]m[L]n

same affinities and multiple sites with different affinities.

possible to define the fraction of occupied binding sites (υ) as:

**3. Binding constants** 

writen as follows:

concentrations.

**3.1 One-site binding** 

$$\text{ID} = \frac{\text{L}\_{\text{bound}}}{\text{P}\_{\text{total}}} = \frac{\text{[PL]}}{\text{[PL]} + \text{[P]}} \tag{6}$$

Determining υ is often easy in spectrophotometric manipulation as will be discussed later. Given υ, equation 5 can then be solved for [PL] and the answer substituted into equation 6 to obtain the quantitative 1:1 stoichiometric model:

$$\mathbf{L}^{\rm D} = \frac{\mathbf{K}\_{\rm u}[\rm L]}{1 + \mathbf{K}\_{\rm u}[\rm L]} \tag{7}$$

This equation is the 1:1 binding isotherm also known as the Langmuir isotherm or the "direct" plot. Its functional form is a rectangular hyperbola whose midpoint will yield Ka. Chemical interpretation of 1:1 binding is that the target P has a single "binding site", as has the ligand L; and when the complex PL forms, no further sites are available for the binding of any additional ligand. To test the 1:1 stoichoimetry equation 7 may be rearrange into a linear plotting form. Since υ is the bound fraction, then 1-υ is the free one: (1- υ) = 1/(1+Ka[L]). Thus υ/(1- υ) = Ka[L], and:

$$
\log \frac{\text{o}}{\text{1} \cdot \text{o}} = \log \text{ [L]} + \log \text{K}\_{\text{a}} \tag{8}
$$

This log-log plot should be linear with a slope of one if the stoichoimetry is really 1:1. This is called a Hill plot. Equation 7 can be also rearranged to three different non-logarithmic linear plotting forms. Taking simply the reciprocal of the equation yields the double-reciprocal plot (used by plotting 1/υ against 1/[L]):

$$\frac{1}{\frac{1}{\alpha}} = \frac{1}{\mathbb{K}\_{\text{a}}[\text{L}]} + 1\tag{9}$$

In spectroscopic studies this plot is commonly known as the Benesi-Hildebrand plot (Benesi & Hildebrand 1949).

Another plot is that of [L]/υ against [L] which is expected to be linear:

$$\frac{[\text{L}]}{\text{o}} = [\text{L}] + \frac{1}{\text{K}\_{\text{a}}} \tag{10}$$

And the third plotting of υ/[L] agains υ, sometimes called Scatchard plot (Scatchard 1949):

$$\frac{1}{\text{m}^3} = \text{tr}\mathbf{K}\_\text{a} + \mathbf{K}\_\text{a} \tag{11}$$

Linearity in all of these plots is a necessary condition if the 1:1 model is valid; and from the parameters of equations Ka can be evaluated. Usually υ is not measured directly but rather some experimental quantity related to it, so that the interpretation of the plots depends on the particular experimental methodology.

#### **3.2 Multi-site binding**

Most biological systems tend to have more than one binding site, that is the case of many systems of small molecules binding to proteins. In these cases we may consider that n ligands may bind to a single target molecule. The average number of ligand molecules bound per target molecule (b) is defined as:

$$\mathbf{D} = \frac{\mathbf{I}\_{\text{total}} - \mathbf{[I.]}}{\mathbf{P}\_{\text{total}}} \tag{12}$$

Assuming that all n binding sites in the target molecule are identical and independent, it is possible to establish:

$$\mathbf{D} = \frac{\text{nk[L]}}{1 + \text{k[L]}} \tag{13}$$

Thermodynamics as a Tool for the Optimization of Drug Binding 771

Plotting log(υ/(1-υ)) against log[L] will yield a straight line with slope nH (called the Hill coefficient). The Hill coefficient is a qualitative measure of the degree of cooperativity and it is experimentally less than the actual number of binding sites in the target molecule. When nH > 1, the system is said to be positively cooperative, while if nH < 1, it is said to be anticooperative. Positively cooperative binding means that once the first ligand is bound to its target molecule the affinity for the next ligand increases, on the other hand the affinity for subsequent ligand binding decreases in negatively cooperative (anti-cooperative) systems. In the case of nH = 1 a non-cooperative binding occurs, here ligand affinity is independent of

Since equation 19 assumes that nH = n, it does not described exactly the real situation. When a Hill plot is constructed over a wide range of ligand concentrations, the continuity of the plot is broken at the extremes concentrations. In fact, the slope at either end is approximately one. This phenomenon can be easily explained: when ligand concentration is either very low or very high, cooperativity does not exist. For low concentrations it is more probable for individual ligands to find a target molecule "empty" rather than to occupy succesive sites on a pre-bound molecule, thus single-binding is happening in this situation. At the other extreme, for high concentrations, every binding-site in the target molecule but one will be filled, thus we find again single-binding situation. The larger the number of sites in a single target molecule is, the wider range of concentrations the Hill plot will show

As discuss above the binding constant provides important and interesting information about the system studied. We will present a few of the multiple experimental posibilities to measure this constant (further information could be found in the literature (Johnson *et al.* 1960; Connors 1987; Hirose 2001; Connors&Mecozzi 2010; Pollard 2010)). It is essencial to keep in mind some crucial details to be sure to calculate the constants properly: it is important to control the temperature, to be sure that the system has reached the equilibrium and to use the correct equilibrium model. One common mistake that should be avoid is

Different techniques are commonly used to study the binding of ligands to their targets. These techniques can be classified as calorimetry, spectroscopy and hydrodynamic methods. Hydrodynamic techniques are tipically separation methodologies such as different chromatographies, ultracentrifugation or equilibrium dialysis with which free ligand, free target and complex are physically separated from each other at equilibrium, thus concentrations of each can be measured. Spectroscopic methodologies include optical spectroscopy (e.g. absorbance, fluorescence), nuclear magnetic resonance or surface plasmon resonance. Calorimetry includes isothermal titration and differential scanning. Calorimetry and spectroscopy methods allow accurately determination of thermodynamics and kinetics

Once the bound (or free) ligand concentration is measured, the binding proportion can be calculated. Other thermodynamic parameters can be calculated by varying ligand or target

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

of the binding, as well as can give information about the structure of binding sites.

confuse the total and free concentrations in the equilibrium expression.

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

whether another ligand is already bound or not.

**4. Determination of binding constants** 

concentrations or the temperature of the system.

cooperativity.

where k is the constant for binding to a single site. According to this equation this system follows the hyperbolic function characteristic for the one-site binding model. To define the model n and k can be evaluated from a Scatchard plot. The affinity constant k is an average over all binding sites, it is in fact constant if all sites are truly identical and independent. A stepwise binding constant (Kst) can be defined which would vary statistically depending on the number of target sites previously occupied. It means that for a target with n sites will be much easier for the first ligand added to find a binding site than it will be for each succesive ligand added. The first ligand would have n sites to choose while the nth one would have just one site to bind. The stepwise binding constant can be defined as:

$$\mathbf{K}\_{\text{st}\mathbf{l}} = \frac{\text{number of free target sites}}{\text{number of bound sites}} \mathbf{k} = \frac{\mathbf{n} - \mathbf{b} + 1}{\mathbf{b}} \mathbf{k} \tag{14}$$

It is interesting to notice that a deviation from linearity in the Scatchard plot (and to a lesser extent in the Benesi-Hildebrand) gives information on the nature of binding sites. A curved plot denotes that the binding sites are not identical and independent.

#### **3.3 Allosteric interactions**

Another common situation in biological systems is the cooperative effect, in that case several identical but dependent binding sites are found in the target molecule. It is important to define the effect of the binding of succesive ligands to the target to describe the system. An useful model for that issue is the Hill plot (Hill 1910). In this case the number of ligands bound per target molecule will be (take into account that the situation in this system for equation 2 is m=1 and n≠1):

$$\text{IP} = \frac{n[\text{PL}\_n]}{[\text{PL}\_n] + [\text{P}]} \tag{15}$$

if equation 5 is solved for [PLn] and substitute into equation 15, then:

$$\mathbf{P} = \frac{\mathbf{n} \mathbb{K}\_{\mathbf{a}} \mathbf{l} \mathbb{L} \mathbb{L}^{n}}{\mathbb{K}\_{\mathbf{a}} \mathbb{L} \mathbb{L}^{n} + 1} \tag{16}$$

This expression can be rewritten as:

$$\frac{\mathbf{h} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} = \mathbf{K}\_{\mathbf{a}} [\mathbf{L}]\_{\mathbf{n}} \tag{17}$$

Note that the fraction of sites bound, υ (see equation 6), is the number of sites occupied, b, divided by the number of sites available, n. Then equation 17 becomes:

$$\frac{1}{\frac{1}{\alpha} \cdot \alpha} = \mathbf{K}\_{\mathbf{a}} [\mathbf{L}]^{\mathbf{n}} \tag{18}$$

Equation 18 is known as the Hill equation. From the Hill equation we arrive at the Hill plot by taking logarithms at both sides:

$$\log \frac{\mathbf{o}}{\mathbf{1} \cdot \mathbf{o}} = \mathbf{n}\_{\text{H}} \log \left[ \mathbf{L} \right] + \log \mathbf{K}\_{\mathbf{a}} \tag{19}$$

Assuming that all n binding sites in the target molecule are identical and independent, it is

b = nk[L]

where k is the constant for binding to a single site. According to this equation this system follows the hyperbolic function characteristic for the one-site binding model. To define the model n and k can be evaluated from a Scatchard plot. The affinity constant k is an average over all binding sites, it is in fact constant if all sites are truly identical and independent. A stepwise binding constant (Kst) can be defined which would vary statistically depending on the number of target sites previously occupied. It means that for a target with n sites will be much easier for the first ligand added to find a binding site than it will be for each succesive ligand added. The first ligand would have n sites to choose while the nth one would have

number of bound sites k = n – b + 1

It is interesting to notice that a deviation from linearity in the Scatchard plot (and to a lesser extent in the Benesi-Hildebrand) gives information on the nature of binding sites. A curved

Another common situation in biological systems is the cooperative effect, in that case several identical but dependent binding sites are found in the target molecule. It is important to define the effect of the binding of succesive ligands to the target to describe the system. An useful model for that issue is the Hill plot (Hill 1910). In this case the number of ligands bound per target molecule will be (take into account that the situation in this system for

b = n[PLn]

b = nKa[L]<sup>n</sup> Ka[L]<sup>n</sup>

Note that the fraction of sites bound, υ (see equation 6), is the number of sites occupied, b,

Equation 18 is known as the Hill equation. From the Hill equation we arrive at the Hill plot

b

υ

just one site to bind. The stepwise binding constant can be defined as:

plot denotes that the binding sites are not identical and independent.

if equation 5 is solved for [PLn] and substitute into equation 15, then:

divided by the number of sites available, n. Then equation 17 becomes:

Kst= number of free target sites

1 + k[L] (13)

<sup>b</sup> <sup>k</sup> (14)

[PLn] + [P] (15)

n - b = Ka[L]<sup>n</sup> (17)

1 - <sup>υ</sup> = Ka[L]<sup>n</sup> (18)

1 - <sup>υ</sup> = nH log [L] + log Ka (19)

+ 1 (16)

possible to establish:

**3.3 Allosteric interactions** 

equation 2 is m=1 and n≠1):

This expression can be rewritten as:

by taking logarithms at both sides:

log <sup>υ</sup>

Plotting log(υ/(1-υ)) against log[L] will yield a straight line with slope nH (called the Hill coefficient). The Hill coefficient is a qualitative measure of the degree of cooperativity and it is experimentally less than the actual number of binding sites in the target molecule. When nH > 1, the system is said to be positively cooperative, while if nH < 1, it is said to be anticooperative. Positively cooperative binding means that once the first ligand is bound to its target molecule the affinity for the next ligand increases, on the other hand the affinity for subsequent ligand binding decreases in negatively cooperative (anti-cooperative) systems. In the case of nH = 1 a non-cooperative binding occurs, here ligand affinity is independent of whether another ligand is already bound or not.

Since equation 19 assumes that nH = n, it does not described exactly the real situation. When a Hill plot is constructed over a wide range of ligand concentrations, the continuity of the plot is broken at the extremes concentrations. In fact, the slope at either end is approximately one. This phenomenon can be easily explained: when ligand concentration is either very low or very high, cooperativity does not exist. For low concentrations it is more probable for individual ligands to find a target molecule "empty" rather than to occupy succesive sites on a pre-bound molecule, thus single-binding is happening in this situation. At the other extreme, for high concentrations, every binding-site in the target molecule but one will be filled, thus we find again single-binding situation. The larger the number of sites in a single target molecule is, the wider range of concentrations the Hill plot will show cooperativity.
