2. Theory of steady-state fluorescence spectroscopy

The use of fluorescence spectroscopy experiments to understand the protein-ligand complex formation has become increasingly frequent in scientific community because many proteins have endogenous fluorescence probe such as tryptophan, tyrosine, and phenylalanine amino acids. Following the fluorescence signal of these probes, it is possible to characterize the molecular mechanism of complex formation [22].

As the name already suggests, this technique makes use of fluorescence as the physical observable. Fluorescence, by definition, is a photon emission mechanism which occurs by the decaying of molecule electrons from a higher energy singlet state to a lower energy singlet state with emission rate of the order of 10<sup>8</sup> s 1 , resulting in a typically fluorescence lifetime close to 10 ns [22].

The fluorescence spectroscopy, when applied in protein-ligand system, consists in analyzing the quenching of fluorescence signal in the presence of different concentrations of ligand. Quenching may be static or collisional (dynamic) depending on the nature of interactions. In collisional quenching, the fluorophore, which is in the excited state, is deactivated returning to its ground state because of diffusive encounters with some other molecule of the solution, called quencher. On the other hand, the static quenching occurs when a fundamental and non-fluorescent complex occurs [22].

In a typical experiment of fluorescence spectroscopy, the way to distinguish the quenching mechanism is analyzing the Stern-Volmer plot (Eq. (1)) in different temperatures, where Fo is the fluorescence intensity in the absence of the quencher and F is the intensity in different concentrations of the quencher, [Lt] [22].

$$\frac{F\_0}{F} = \mathbf{1} + K.[L\_t] \tag{1}$$

The increase in the constant K as a consequence of the increase in temperature is a strong indication that the quenching mechanism is collisional; in that case the constant is usually denoted by Kd. On the other hand, if the constant K decreases with the increasing in temperature, the characteristic mechanism is the static one, and the constant is usually denoted by Ksv. Figure 3 shows the different behaviors of the static and collisional quenching [22].

If the mechanism determined was the static one, there are couples of models that can be applied in the system in order to obtain the association constant Ka or also called binding constant. The most common and simplest model reported in literature to study protein-ligand is the binding equilibria for a first-order reaction, where the ligands (L) are entering one by one at the binding site of the protein (P) [23].

$$\mathbf{P} + \mathbf{L} = \mathbf{P} \mathbf{L} \rightarrow \mathbf{L} \mathbf{P} + \mathbf{L} = L\_2 \mathbf{P} \rightarrow L\_2 \mathbf{P} + \mathbf{L} = L\_3 \mathbf{P} \dots \rightarrow L\_{n-1} \mathbf{P} + L = L\_n \mathbf{P}$$

Assuming that the sites are equivalents and independents, the association constant is

$$K\_a^n = \frac{[L\_n.P]}{[L]^n.[P]} \tag{2}$$

where [P] is the free protein concentration, [L] is the free quencher concentration, and [LnP] is the concentration of protein-quencher complex. In this model, the protein is either free in solution or bound to quencher, then

$$[P\_t] = [L\_nP] + [P] \tag{3}$$

where [Pt] is the concentration of total protein in the system [22]. Considering that the fluorescence intensity is only emitted by free proteins

Figure 3. Examples of different quenching mechanisms. (A) Collisional quenching and (B) static quenching.

Molecular Mechanism of Flavonoids Using Fluorescence Spectroscopy and Computational Tools DOI: http://dx.doi.org/10.5772/intechopen.84480

$$\frac{[P\_t]}{[P]} = \frac{F\_0}{F} \tag{4}$$

Eq. (2) can be rearranged to

$$K\_a^n = \left(\frac{F\_0 - F}{F}\right) \cdot \left(\frac{1}{[L\_t] - \left(\frac{F\_0 - F}{F}\right) . [P\_t]}\right)^n \tag{5}$$

Applying the logarithm function in Eq. (5) and rearranging it

$$\log\left(\frac{F\_0 - F}{F}\right) = n. \log K\_d - n. \log\left(\frac{1}{[L\_t] - \left(\frac{F\_0 - F}{F}\right). [P\_t]}\right) \tag{6}$$

Eq. (6) is known as double-logarithm equation, where one can calculate the binding constant Ka. The binding constant is related to thermodynamic parameter through the van't Hoff Equation [24]:

$$
\ln K\_a = -\frac{\Delta H}{R.T} + \frac{\Delta S}{R} \tag{7}
$$

where R is the universal gas constant (≈ 8.31 J/mol.K) and T is the temperature. In a graphic of ln Ka versus T, the enthalpy change (ΔH) and the entropy change (ΔS) are obtained by the slope of the linear function and the linear coefficient, respectively.

Ross and Subramanian [25] associated the enthalpy variation and entropy variation with the most predominant interactions that stabilize the complex, as showed in Table 1.

Besides that, Gibbs free energy variation says about the spontaneity of the complex formation process, where ΔG > 0 indicates a non-spontaneity process and ΔG < 0 indicates a spontaneity process. The enthalpy variation, on the other hand, indicates if the system is either an exothermic process (ΔH < 0) or an endothermic process (ΔH > 0) [24, 25].

Unlike the binding equilibrium method that is based upon a first-order reaction predicting a single binding site, there is another method that does not make use of a previous model, and in addition to determining the number of binding sites, it can still determine the cooperativity, if any. This method was developed by Scatchard [26], and it is not as popular as binding equilibrium method to characterize proteinligand interaction in the scientific community yet.

This method makes use of another technique to obtain preliminary data known as binding density function (BDF) [27], where a physical observable is chosen to follow the interactions, such as absorbance and fluorescence intensity, among others. Supposing that the intensity of fluorescence emitted by the protein was chosen as physical observable, in which in an aqueous system containing protein is titrated by the ligand. Considering the system in equilibrium, the average number of quencher bound by protein Σν<sup>i</sup> is determined from a given free quencher


Table 1. Expected signs of contributions to ΔH and ΔS. concentration (L); in this context, if the free quencher concentration is the same for two or more solutions of different concentrations of total protein Pt, then the average distribution of the binding density of the quencher will also be the same, reaching in the expression for the mass conservation [27]:

$$[L\_t] = [L] + (\sum \nu\_i) . [P\_t] \tag{8}$$

where [Lt] is the total quencher concentration, [L] is the free quencher concentration, and [Pt] is the total protein concentration. In order to obtain the average number of quenchers bound (Σνi), a graphic of total quencher concentration [Lt] versus the total protein [Pt] can be plotted, in which by the angular coefficient, the Σν<sup>i</sup> is obtained and by the linear coefficient of the linear function [L] is obtained. The values for [Pt] and [Lt] are obtained from the graphic of ΔF versus log [Lt] shown in Figure 4, where ΔF is the suppression percentage [27].

Once the values of Σν<sup>i</sup> and [L] were obtained from the graphic in the BDF method, the method developed by Scatchard [26] can be used to follow proteinligand interaction. To determine all those information from the method, a graphic of Σνi/[L] versus Σν<sup>i</sup> is plotted, and from the function obtained in the graphic, it can be said that the sites have cooperativity or they are equivalents and independents.

#### Figure 4.

Example of graph used in BDF theory. In this case, the percentage of quenching was measured in two different concentrations of protein, PT1 and PT2.

Figure 5. Illustration of two Scatchard graphics. (A) Negative cooperativity and (B) positive cooperativity.

Molecular Mechanism of Flavonoids Using Fluorescence Spectroscopy and Computational Tools DOI: http://dx.doi.org/10.5772/intechopen.84480

Bordbar and co-workers [28] reported the mathematical functions that describe the behavior of the system in the Scatchard plots with the type of cooperativity. They determined that if the function is polynomial with the positively concavity (Figure 5(A)), the protein has negative cooperativity between the sites. If instead there is a polynomial function with negative concavity, the protein has a positive cooperativity among the sites (Figure 5(B)) [28].

For these cases where the Scatchard graph shows a cooperativity function shape, one can apply the equation developed by Hill [24] to determine the number of sites n of each set of equal sites and the Ka association constant of each set of interaction sites:

$$\sum \nu\_i = \frac{n\_{1\cdot}(K\_{a1\cdot}[L])^{H1}}{\mathbf{1} + (K\_{a1\cdot}[L])} + \frac{n\_{2\cdot}(K\_{a2\cdot}[L])^{H2}}{\mathbf{1} + (K\_{a2\cdot}[L])} \tag{9}$$

where [L] is the concentration of free quenchers and H is Hill's index. If H = 1 the system is noncooperative, H > 1 the system has positive cooperativity, and H < 1 the system has negative cooperativity [24, 28].

In the case that the function, which describes the system, is linear, the protein has no cooperativity among the sites, and all sites are identical and independent. For this behavior, Scatchard [26] developed his own mathematical model to find the number of sites n and the association constant Ka:

$$
\Sigma \nu\_i = \frac{n.K\_a.[L]}{1 + K\_a.[L]} \tag{10}
$$

To rearrange in the form of linear equation to model the Scatchard graphic, it is given as follows:

$$\frac{\sum \nu\_i}{[L]} = n.K\_a - K\_a. \sum \nu\_i \tag{11}$$

## 3. Theory of molecular docking

Molecular docking is a powerful technique that has been used in association with experimental data to determine ligand binding sites in targets with pharmacological interest. It can also predict the ligand conformation in the binding site and consequently the interactions that stabilize the complex. In addition, molecular docking has been also used as an efficient and a cheap technique for virtual screening large molecule databases and selects compounds which bind with specificity in pharmaceutical targets before carrying out in vitro and in vivo experiments [29, 30].

Molecular docking consists in determining the most probable conformations of the complex composed by receptor ligand based on an energy ranking of each conformation. To obtain this energy ranking, the ligand is put to interact in an environment under a protein force field including non-covalent potentials such as van der Waals force, hydrogen bonds, and electrostatic nature forces [29]. There are several softwares to perform this kind of prediction such as AutoDock, GOLD, DOCK, and FlexX, just to mention a few.

Each of those mentioned software has its own potential mathematical function to elucidate the forces involved in the complex and calculate the energy score to be ranked. The following is an example of potential utilized by AutoDock04 [31] software:

$$\mathbf{V} = I. \sum\_{\vec{i}\vec{j}} \left( \frac{\mathbf{A}\_{\vec{i}\vec{j}}}{r\_{\vec{i}\vec{j}}^{12}} - \frac{\mathbf{B}\_{\vec{i}\vec{j}}}{r\_{\vec{i}\vec{j}}^{6}} \right) + J. \sum\_{\vec{i}\vec{j}} E(\mathbf{t}). \left( \frac{\mathbf{C}\_{\vec{i}\vec{j}}}{r\_{\vec{i}\vec{j}}^{12}} - \frac{D\_{\vec{i}\vec{j}}}{r\_{\vec{i}\vec{j}}^{10}} \right) + K. \sum\_{\vec{i}\vec{j}} \frac{q\_{\vec{i}} \cdot q\_{\vec{j}}}{e.r\_{\vec{i}\vec{j}}^{2}} + \Delta W\_{S} \tag{12}$$

The weighting constants I, J, E(t), K, and W are those optimized to calibrate the empirical free energy based on a set of experimentally characterized complexes. The first term is the Lenard-Jones potential, in which parameters A and B are taken from the Amber force field [31]. The second term refers to the hydrogen bond in which the parameters C and D are obtained to ensure a minimum energy of 5 kcal/mol in 1.9 Å for O-H and N-H and 1 kcal/mol in 2.5 Å for S-H. The function E(t) provides directionality based on the angle t of the geometry of an ideal hydrogen bond [31]. The third term is a shielded Coulomb potential for electrostatic interaction. The last term is the desolvation potential based on the volume of the atoms surrounding a given atom and shelter it of the solvent [31].

To reduce the computational costs, the programs usually make use of a precalculation type. The software creates grid maps for each type of atom present in the ligand to be used in the docking. Grid maps consist of a three-dimensional array of regularly spaced points centered on the active site of the protein or macromolecule under study. Each point inside the grid maps records the energy interaction of a test atom with the protein [31]. The complexity of finding the best conformation requires computational methods with the potential to effectively investigate a large number of possible solutions, aiming to find the best result. The search algorithms that are usually utilized by molecular docking softwares can be classified into three categories such as systematic, deterministic, and stochastic search methods [32].

In systematic search algorithms, each degree of freedom has a set of values, so that all degrees of freedom of the molecule are explored combinatorically during the search. Examples of systematic search algorithms are anchor-and-grow or incremental construction algorithm [32]. The deterministic search methods are characterized by the fact that, given the same initial input state, they always produce the same output. It happens because the initial state determines the possible movement to generate the next state, which has to have the same or lower energy than the previous state. Examples of deterministic search are energy minimization methods and molecular dynamics (MD) simulations [32]. Stochastic methods vary randomly in all degrees of freedom of the ligand (translational, rotational, and conformational) at each step, generating a great diversity of solutions. The solutions are evaluated according to a probabilistic criterion to decide if they will be accepted or not. In this way this method requires a large number of conformations to obtain a desired result. Monte Carlo, simulated annealing, and evolutionary methods are examples of stochastic search method, and they are the most common in molecular docking softwares [32, 33].
