**2. Principles**

2 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

bottlenecks of methods employed in such process, and the directions for future

The information content provided by thermodynamic parameters is vast. It plays a prominent role in the elucidation of the molecular mechanism of the binding phenomenon, and – through the link to structural data – enables the establishment of the structure-activity relationships, which may eventually lead to rational design. However, the deconvolution of the thermodynamic data and particular contributions is not a straightforward process; in

Two groups of computational methods, which are particularly useful in assessment of the thermodynamics of molecular recognition events, will be discussed. One of them are methods based on molecular dynamics (MD) simulations, provide detailed insights into the nature of ligand-protein interactions by representing the interacting species as a conformational ensemble that follows the laws of statistical thermodynamics. As such, these are very valuable tools in the assessment of the dynamics of such complexes on short (typically, picosecond to tens of nanosecond, occasionally microsecond) time scales. I will give an overview of free energy perturbation (FEP) methods, thermodynamic integration (TI), and enhanced sampling techniques. The second group of computational methods relies on very accurate determinations of energies of the macromolecular systems studied, employing calculations based on approximate solutions of the Schrödinger equation. The spectrum of these quantum chemical (QM) methods applied to study ligand-protein interactions is vast, containing high-level ab initio calculations: from Hartree-Fock, through perturbational calculations, to coupled-clusters methods; DFT and methods based on it (including "frozen" DFT and SCC-DFTTB tight binding approaches); to semi-empirical Hamiltonians (such as AM1, PM3, PM6, just to mention the most popular ones) (Piela, 2007, Stewart, 2009). Computational schemes based the hybrid quantum mechanical –molecular mechanical (QM/MM) regimes will also be introduced. Due to the strong dependence of the molecular dynamics simulations on the applied force field, and due to the dependence of both MD simulations and QM calculations on the correct structure of the complex, validation of results obtained by these methodologies against experimental data is crucial. Isothermal titration calorimetry (ITC) is one of the techniques commonly used in such validations. This technique allows for the direct measurement of all components of the Gibbs' equation simultaneously, at a given temperature, thus obtaining information on all the components of free binding energy during a single experiment. Yet since these are *de facto* global parameters, the decomposition of the factors driving the association, and investigation of the origin of force that drives the binding is usually of limited value. Nonetheless, the ITC remains the primary tool for description of the thermodynamics of ligand-protein binding (Perozzo *et al*., 2004). In this chapter, I will give a brief overview of ITC and its applicability in the description of recognition events and to molecular design. Another experimental technique, which has proven very useful in the experimental validation of computational results, is NMR relaxation. These measurements are extremely valuable, as they specifically investigate protein dynamics on the same time scales as MD simulations. As such, the results obtained can be directly compared with simulation outputs. In addition, the Lipari-Szabo model-free formalism (Lipari and Szabo, 1982) is relatively free of assumptions regarding the physical model describing the molecular motions. The only requirement is the internal dynamics being uncorrelated with the global tumbling of the system under investigation. The results of the Lipari–Szabo analysis, in the form of generalised order parameters ( <sup>2</sup> *SLS* ), can be readily interpreted in terms of the

particular, assessing the entropic contributions is often very challenging.

development.

## **2.1 Enthalpic and entropic components of free binding energy**

A non-covalent association of two macromolecules is governed by general thermodynamics. Similarly to any other binding event (or – in a broader context – to any spontaneous process), it occurs only when it is coupled with a negative Gibbs' binding free energy (1), which is the sum of an enthalpic, and an entropic, terms:

$$
\Delta \mathbf{G} = \Delta \mathbf{H} - \mathbf{T} \, \Delta \mathbf{S} \tag{1}
$$

*G GG G obs i sb s* 

The equation above shows how the observable free energy of binding can be decomposed into the 'intrinsic' term, and the solvation contributions from the ligand-protein complex and unbound interactors. Similar decomposition can be done for the enthalpic and entropic

Since the enthalpic and entropic contributions to the binding free energy depend on many system-specific properties (such as protonation states, binding of metal cations, changes in conformational entropy from one ligand to another in a way which is very difficult to predict, etc), the conclusion is that optimising the overall free energy remains the most viable approach to rational (structure-based) molecular design. Attempting to get an insight into individual components of the free energy requires re-thinking the whole concept of ligand-protein binding. This means regarding ligand-protein complexes as specifically interacting yet flexible ensembles of structures rather than rigid entities, and the role of solvation effects. The significant contribution of specific interactions and flexibility to the 'intrinsic' component of binding free energy, and solvation effects will be discussed next in

Electrostatic interactions, involved in ligand-protein binding events, can be roughly classified into three types; charge-charge, charge-dipole, and dipole-dipole. Typical chargecharge interactions are those between oppositely charged atoms, ligand functional groups, or protein side chains, such as positively charged (amine or imine groups, lysine, arginine, histidine) and negatively charged (carboxyl group, phosphate groups, glutamate side chain). An important contribution to the enthalpy change associated with a binding event arises from charge-dipole interactions, which are the interactions between ionised amino acid side chains and the dipole of the ligand moiety or water molecule. The dipole moments of the

Van der Waals interactions are very important for the structure and interactions of biological molecules. There are both attractive and repulsive van der Waals interactions that control binding events. Attractive van der Waals interactions involve two induced dipoles that arise from fluctuations in the charge densities that occur between adjacent uncharged atoms, which are not covalently bound. Repulsive van der Waals interactions occur when the distance between two involved atoms becomes very small, but no dipoles are induced. In the latter case, the repulsion is a result of the electron-electron repulsion that occurs in

Van der Waals interactions are very weak (0.1- 4 kJ/mol) compared to covalent bonds or electrostatic interactions. Yet the large number of these interactions that occur upon molecular recognition events makes their contribution to the total free energy significant. Van der Waals interactions are usually treated as a simple sum of pairwise interatomic interactions (Wang *et al.*, 2004). Multi-atom VdW interactions are, in most cases, neglected. This follows the Axilrod-Teller theory, which predicts a dramatic (i.e. much stronger than for pairwise interactions) decrease of three-atom interactions with distance (Axilrod and Teller, 1943). Indeed, detailed calculations of single-atom liquids (Sadus, 1998) and solids

polar side chains of amino acid also affect their interaction with ligands.

 

*<sup>f</sup>* (3)

terms separately, as these terms are also state functions.

this chapter.

**2.2 Specific interactions 2.2.1 Electrostatic interactions** 

**2.2.2 Van der Waals interactions** 

two partly-overlapping electron clouds.

where *G is free binding energy, H is enthalpy, S entropy, and T is the temperature.* The enthalpic contribution to the free energy reflects the specificity and strength of the interactions between both partners. These include ionic, halogen, and hydrogen bonds, electrostatic (Coulomb) and van der Waals interactions, and polarisation of the interacting groups, among others. The simplest description of entropic contribution is that it is a measure of dynamics of the overall system. Changes in the binding entropy reflect loss of motion caused by changes in translational and rotational degrees of freedom of the interacting partners. On the other hand, changes in conformational entropy may be favourable and in some cases these may reduce the entropic cost of binding (MacRaild *et al*., 2007). Solvation effects, such as solvent re-organisation, or the release of tightly bound water upon ligand binding can contribute significantly to the entropic term of the binding free energy.

The Gibbs equation can be also written as in equation (2):

$$
\Delta \mathbf{G} = -RT \ln \mathbf{K}\_d \tag{2}
$$

where R is a gas constant, T is the temperature, and *Kd* is binding constant. This formulation emphasises the relationship between Gibbs energy and binding affinity. The ligand-protein association process can be represented in the form of a Born-Haber cycle. A typical cycle is showed in Figure 1. The 'intrinsic' free energy of binding between ligand L and protein P is represented by *Gi* , whereas the experimentally observable free energy of binding is represented by *Gobs .* 

Fig. 1. An example of Born-Haber cycle for ligand-protein (LP) association. It relates the experimentally observed free energy of binding ( *Gobs* ) with 'intrinsic' free energy of binding ( *Gi* ) between ligand (L) and protein (P) and with solvation free energies of free interactors ( *Gsf* ) and the resulting complex ( *Gsb* ). X, Y, Z, and B refer to the number of water molecules involved in solvation of the unbound ligand (X), unbound protein (Y), ligand-protein complex (Z), and to the bulk solvent (B).

Two additional processes can be defined: the free energy of solvation of the free (unbound) interacting partners ( *Gsf* ), and the free energy of solvation of the ligand-protein complex ( *Gsb ).* Since the free energy is a state function, it is independent of the path leading from from one state of the system to another. Hence, the observable free energy of binding can be written as in equation (3):

$$
\Delta \mathbf{G}\_{\rm obs} = \Delta \mathbf{G}\_i + \Delta \mathbf{G}\_{\rm sb} - \Delta \mathbf{G}\_{\rm sf} \tag{3}
$$

The equation above shows how the observable free energy of binding can be decomposed into the 'intrinsic' term, and the solvation contributions from the ligand-protein complex and unbound interactors. Similar decomposition can be done for the enthalpic and entropic terms separately, as these terms are also state functions.

Since the enthalpic and entropic contributions to the binding free energy depend on many system-specific properties (such as protonation states, binding of metal cations, changes in conformational entropy from one ligand to another in a way which is very difficult to predict, etc), the conclusion is that optimising the overall free energy remains the most viable approach to rational (structure-based) molecular design. Attempting to get an insight into individual components of the free energy requires re-thinking the whole concept of ligand-protein binding. This means regarding ligand-protein complexes as specifically interacting yet flexible ensembles of structures rather than rigid entities, and the role of solvation effects. The significant contribution of specific interactions and flexibility to the 'intrinsic' component of binding free energy, and solvation effects will be discussed next in this chapter.

### **2.2 Specific interactions**

4 Thermodynamics – Interaction Studies – Solids, Liquids and Gases

*G H TS* 

The enthalpic contribution to the free energy reflects the specificity and strength of the interactions between both partners. These include ionic, halogen, and hydrogen bonds, electrostatic (Coulomb) and van der Waals interactions, and polarisation of the interacting groups, among others. The simplest description of entropic contribution is that it is a measure of dynamics of the overall system. Changes in the binding entropy reflect loss of motion caused by changes in translational and rotational degrees of freedom of the interacting partners. On the other hand, changes in conformational entropy may be favourable and in some cases these may reduce the entropic cost of binding (MacRaild *et al*., 2007). Solvation effects, such as solvent re-organisation, or the release of tightly bound water upon ligand binding can contribute significantly to the entropic term of the binding free

*H is enthalpy, S*

 

(1)

 *entropy, and T is the temperature.*

*G RT K* ln *<sup>d</sup>* (2)

*Gobs* ) with 'intrinsic' free energy of

*Gsb* ). X, Y, Z, and B refer to the number of

where R is a gas constant, T is the temperature, and *Kd* is binding constant. This formulation emphasises the relationship between Gibbs energy and binding affinity. The ligand-protein association process can be represented in the form of a Born-Haber cycle. A typical cycle is showed in Figure 1. The 'intrinsic' free energy of binding between ligand L and protein P is

Fig. 1. An example of Born-Haber cycle for ligand-protein (LP) association. It relates the

water molecules involved in solvation of the unbound ligand (X), unbound protein (Y),

*Gi* ) between ligand (L) and protein (P) and with solvation free energies of free

*Gsf* ), and the free energy of solvation of the ligand-protein complex

Two additional processes can be defined: the free energy of solvation of the free (unbound)

*Gsb ).* Since the free energy is a state function, it is independent of the path leading from from one state of the system to another. Hence, the observable free energy of binding can be

*Gi* , whereas the experimentally observable free energy of binding is

The Gibbs equation can be also written as in equation (2):

where

energy.

represented by

represented by

binding (

( 

interactors (

interacting partners (

written as in equation (3):

experimentally observed free energy of binding (

*Gsf* ) and the resulting complex (

ligand-protein complex (Z), and to the bulk solvent (B).

*Gobs .* 

*G is free binding energy,* 

#### **2.2.1 Electrostatic interactions**

Electrostatic interactions, involved in ligand-protein binding events, can be roughly classified into three types; charge-charge, charge-dipole, and dipole-dipole. Typical chargecharge interactions are those between oppositely charged atoms, ligand functional groups, or protein side chains, such as positively charged (amine or imine groups, lysine, arginine, histidine) and negatively charged (carboxyl group, phosphate groups, glutamate side chain). An important contribution to the enthalpy change associated with a binding event arises from charge-dipole interactions, which are the interactions between ionised amino acid side chains and the dipole of the ligand moiety or water molecule. The dipole moments of the polar side chains of amino acid also affect their interaction with ligands.

#### **2.2.2 Van der Waals interactions**

Van der Waals interactions are very important for the structure and interactions of biological molecules. There are both attractive and repulsive van der Waals interactions that control binding events. Attractive van der Waals interactions involve two induced dipoles that arise from fluctuations in the charge densities that occur between adjacent uncharged atoms, which are not covalently bound. Repulsive van der Waals interactions occur when the distance between two involved atoms becomes very small, but no dipoles are induced. In the latter case, the repulsion is a result of the electron-electron repulsion that occurs in two partly-overlapping electron clouds.

Van der Waals interactions are very weak (0.1- 4 kJ/mol) compared to covalent bonds or electrostatic interactions. Yet the large number of these interactions that occur upon molecular recognition events makes their contribution to the total free energy significant.

Van der Waals interactions are usually treated as a simple sum of pairwise interatomic interactions (Wang *et al.*, 2004). Multi-atom VdW interactions are, in most cases, neglected. This follows the Axilrod-Teller theory, which predicts a dramatic (i.e. much stronger than for pairwise interactions) decrease of three-atom interactions with distance (Axilrod and Teller, 1943). Indeed, detailed calculations of single-atom liquids (Sadus, 1998) and solids

examined, as it is likely to vary considerably from one ligand-protein system to another one

Regarding weak hydrogen bonds, the most prominent donor is the CH group. These interactions, despite of their weakness, play an important role in stabilising appropriate conformations of ligand-protein complexes, for instance among the complexes between protein kinases and their inhibitors (Bissantz *et al.*, 2010). Protonated histidines can also act as strong CH donors (Chakrabarti and Bhattacharyya, 2007). Weak hydrogen bonds, their nature, and their role in ligand-protein interactions have been extensively reviewed by

The concept of halogen bonds is similar to hydrogen bonds: both types of interactions involve relationships between an electron donor and electron acceptor. In hydrogen bonding, a hydrogen atom acts as the electron acceptor and forms a non-covalent bond by accepting electron density from an electronegative atom ("donor"). In halogen bonding, a

Despite of their prevalence in complexes between proteins and small organic inhibitors (many of them contain halogen atoms due to solubility and bioavailability) and their importance for medicinal chemistry, the significance of halogen bonds in biological context has been overlooked for a long time (Zhou *et al.*, 2010). For a number of years, halogen atoms were regarded as hydrophobic appendages, convenient – from the molecular design point of view - to fill apolar protein cavities. The nature of halogen interactions (such as directionality, sigma-holes) was not studied in detail and not regarded as very important. Indeed, halogen bonds are, in general, fairly weak interactions. On the other hand, in some cases they can compete with hydrogen bonds, thus should be considered in more details, given the importance of hydrogen bonds in ligand-protein interactions and given that many of synthesised small organic compounds contain halogen bonds in their structure (Bissantz

Halogens involved in halogen bonds are chlorine, bromine, iodine, and fluorine (not very often). All four halogens are capable of acting as donors (as proven by computational and experimental data) and follow the general trend: F < Cl < Br < I, with iodine normally forming the strongest bonds, as the strength increases with the size of the halogen atom. From the chemical point of view, the halogens, with the exception of fluorine, have unique electronic properties when bound to aryl or electron withdrawing alkyl groups. They show an anisotropy of electron density distribution with a positive area (so-called -hole) of electrostatic potential opposite the carbon-halogen bond (Clark *et al.*, 2007). The molecular origin of the -hole can be explained quantum chemically and the detailed description is provided in the work by Clark and coworkers (2007). Briefly, a patch of negative charge is formed around the central region of the bond between carbon and halogen atom, leaving the

Available experimental data show the strong influence of halogen bonds on binding affinity. Replacement of hydrogen by halogen atom is often used by medicinal chemists in order to increase the affinity. Indeed, in a series of adenosine kinase inhibitors, a 200-fold affinity gain from hydrogen to iodine has been observed (Iltzsch *et al.*, 1995). Another spectacular, 300-fold affinity difference upon iodine substitution was observed in a series of HIV reversetranscriptase inhibitors (Benjahad *et al.*, 2003). Unsurprisingly, substitution of hydrogen by iodine typically leads to the largest affinity gain, since the strength of the halogen bond

(Barratt *et al.*, 2005, 2006).

Panigrahi and Desiraju (2007).

halogen atom is the donor.

*et al.*, 2010, Zhou *et al.*, 2010).

outermost region positive (hence the "hole").

increases with the size of halogen atom.

**2.2.4 Halogen bonds and multipolar interactions** 

(Donchev, 2006) indicate that multi-body effects amount to only 5% of the total energy (Finkelstein, 2007). However, Finkelstein (2010) shows that those largely ignored multi-atom Van der Waals interactions may lead to significant changes in free energy in the presence of covalent bonds. Those changes can be comparable to those caused by the substitutions of one atom by another one in conventional pairwise Van der Waals interactions. Thus, the currently used force fields (applied in MD simulations) need to be revised.
