**2. Context and relevant aspects**

instance, of phase diagrams, protein-ligand and drug binding affinities, drug partitioning, reaction rates, equilibrium constants, acid-base equilibria, solvation contributions, or confinement energies. The prediction of these quantities, with relevance in the field of computeraided drug design requires knowledge of the related free-energy changes [2]. Most of the research topics involving molecular modeling and simulation, with free-energy calculations, stem from pharmaceutical applications [1, 3, 4]. Those methods have provided insight into the time evolution of host-guest systems and of protein-ligand interactions [2, 4] and have revealed the respective dynamic behavior in water, thus establishing relevant recognition and affinity patterns. However, the formation of inclusion complexes is frequently used as a drug solubilizing approach, aiming at better therapeutic outcomes. This makes results sparse and focused on individual systems, lacking a comprehensive characterization for assessing the factors that govern, for example, the inclusion process. Despite the continuous progress and development of improved algorithms, the estimation of binding free energies by molecular dynamics (MD) simulations still requires significant computational efforts due to the mathematical complexity imposed by the solvated systems, often composed by myriads of atoms [3]. Recently, Pais and co-workers [5] have proposed an automated procedure based on umbrella sampling and the "flexible molecule" approximation for the calculation of binding constants in complexes formed between β-cyclodextrin and several naphthalene derivatives. In this type of complexes, the guest molecule may alter the structure of the host, leading to relevant cooperative effects, when compared to the free molecules. Inclusion of guests into host molecules is essentially governed by (i) relatively weak non-covalent interactions (NCI), including hydrophobic, van der Waals and electrostatic interactions, π-π stacking, hydrogen bonding, or a combination of these interactions, and also by (ii) stronger ionic and dipolar interactions. In addition to several reviews (see e.g. [6, 7]), including many dealing with specific host-guest systems [4, 7–12], a large number of research papers are available on this subject, including some focused on mechanistic aspects of the inclusion phenomena (see e.g. Refs. [5, 13–15].)

8 Molecular Dynamics

In fact, both the efficiency and accuracy of free-energy calculations have been greatly improved by a wide variety of methods [16]. However, it is increasingly difficult for researchers to find their way through the maze of available computational techniques. Why are there so many methods? Are they conceptually related? Do they differ in efficiency and accuracy? Why do methods that appear to be very similar carry different names? Which method is the best for a specific problem? How to choose the most relevant method to tackle the system at hand? These questions may leave researchers in the field confused and looking for clear guidelines. However, answers to these questions are not straightforward. A distinction has been made between two classes of free-energy transformations, namely those of alchemical and geometrical nature [17]. While the former exploits the malleability of the potential energy function and the virtually infinite possibilities of computer simulations to transform between chemically distinct states, the latter includes positional, orientational, and conformational changes in macromolecules and complexes thereof. At the practical level, such transformations are achieved using a variety of approaches, which can be separated into four main groups, (i) probability distributions and histograms, (ii) perturbation theory, (iii) non-equilibrium work and (iv) gradient-based methods [18]. One recurrent question concerns the selection of the most relevant approach to tackle the problem at hand, which can be reworded in terms of the best-suited, cost-effective method to obtain a reliable answer. To a large extent, this question In the context of non-covalent binding, host-guest systems have emerged as useful models for assessing the accuracy of simulation methods. This can be explained by the fact that these systems display interesting features comparable to protein-ligand binding, including hydrogen bonds, conformational restriction and desolvation, and also possess an adequate size for conducting more precise thermodynamic calculations [24]. This is the case of cyclodextrin host-guest systems, which mimic several characteristics of protein-ligand binding with the advantage of being much more accessible due to their small size. These characteristics also facilitate the selection of the force field (FF) based on the attribution of the level of accuracy.

Specifically, the combination of MD and free-energy calculations with experimental results have revealed the relevant interaction patterns in the formation of inclusion complexes between several host (e.g. cyclodextrins [4, 5], cucurbiturils [25] and bambusurils [8, 26]) and guest molecules (e.g. antitumor and antiviral drugs [27], steroids [28], flavonoids [29] and small ionic species [26]). These small aggregates, comprising typically two (in 1:1 host-guest complexes) or three molecules (in 1:2 complexes) have been the basis of fundamental developments for establishing what governs associations in soft-matter and stability in larger supramolecular nanostructures, such as nanogels and targeted nanoparticles.

Despite significant advances in molecular modeling techniques and the comparative simplicity of host-guest systems, there is still a need for a tractable and theoretically sound computational method to interpret experimental data and help with the design of new hosts for targeted molecular guests. Similarly to other macromolecular systems, host-guest systems exhibit marked entropy-enthalpy compensation. Such property has been observed for NCI in both water and organic solvents. Among the wide applications of alchemical transformations, host-guest chemistry occupies the most prominent position [30].

A fundamental aspect to be understood is the precise manner in which the guest molecule binds to the host. MD simulations have greatly contributed to understanding molecular binding phenomena as a fully dynamical process. However, these simulations are limited by the time scales that can be routinely sampled. The introduction of special-purpose machines and the evolution of parallel codes, has increased enormously the time scales accessible by fully atomistic MD. However, guest molecules (e.g. drugs) with long residence times are common, and the respective association/dissociation from a host or receptor cannot be observed by conventional MD calculations even when specialized hardware is employed. This well-recognized limitation of MD has led to the development of various algorithms to enhance the sampling of the high free-energy states and rare events, associated to high free-energy barriers.

The ability to decouple relevant energy contributions from individual components is also far from a simple solution, being the basis of alchemical free-energy methods [19, 20], such as the thermodynamic integration (TI) [17, 44, 45], free-energy perturbation (FEP) [20] and molecu-

Modeling Soft Supramolecular Nanostructures by Molecular Simulations

http://dx.doi.org/10.5772/intechopen.74939

11

The feasibility of the estimated binding free-energy depends on the correct estimation of thermodynamic quantities and the concerted interaction components (enthalpy, entropy and solvent contributions) [15, 47]. For instance, the quantification of these interaction components have been directed at inclusion complexes between cyclodextrins and its derivatives and different model drugs (see Refs. [13, 15, 48].). However, only a few studies (e.g. [5, 13]) have introduced technical and practical directions to investigate the energy components underlying the stability of these complexes. Spoel and co-workers [13] have emphasized the role of solvent contribution resorting to steered molecular dynamics and PMF calculations. The latter techniques have also been used by Pais and co-workers [5] for assessing the relevance of non-included guest moieties in the binding constants of cyclodextrin inclusion complexes, estimating the weight of enthalpic and entropic contributions to the free-energy. In this context, MD and PMF calculations allow the modulation of the formation processes in inclusion complexes, along which the free-energy profiles are derived from an umbrella sampling procedure [5]. This approach has been employed in the characterization of the inclusion of a wide range of drugs on cyclodextrins, and also in the aggregation of host units either in dimerization [15, 48] or in rotaxanes formation [49, 50]. Other hybrid approaches [51], including quantum mechanics/molecular mechanics (QM/MM) [29] methods have also provided insight on the formation of inclusion com-

**3. Using potentials of mean force to address host-guest recognition** 

Several research groups have opened the way for future progress through innovative applications of free-energy methods to physical and organic chemistry, as well as structural biology. An exhaustive account of the wide range of works published in the early years of free-energy calculations falls beyond the scope of this section. The reader is referred to Refs. [16, 52] for a

A complete thermodynamic characterization of the binding process implies the knowledge of the enthalpy and entropy of association. This is one of the key elements in identifying the stabilizing factors and in understanding how is the guest and host assemble. Standard molecular simulation methods have reproduced the absolute thermodynamic properties of binding (standard free-energy, enthalpy and entropy) between host-guest systems [53]. The absolute binding free-energy can be expressed as the sum of separate free-energy contributions corresponding to a step-by-step process describing the association process between the host and guest [5]. This can be accomplished by calculating the PMF profile along a specific reaction coordinate characterizing the association process (this reaction coordinate can be the

distance between the center of mass of the host and that of the guest).

lar mechanics-Poisson-Boltzmann surface area (MM-PBSA) [46].

plexes at a molecular level.

full description of these efforts.

**and association**

Enhanced sampling methods can speed up conformational sampling by various means and represent an effective alternative to access with high accuracy the thermodynamics and possibly the kinetics that underlie these processes. Steered MD and umbrella sampling [31], with potential of mean force estimation (PMF) [32] allows inspecting the free-energy profile and the mechanistic aspects involved in formation of host-guest complexes [5].

In recent years, other strategies aimed at the same goal have been proposed. Replica exchange [33] metadynamics [34, 35], accelerated MD [36], milestoning [37], transition path sampling [38], and combinations of these are among the most widely used methods to enhance conformational sampling. For instance, the replica exchange methodology has been used in the atomistic simulations, as well as in a number of coarse-grained simulations (see e.g. [34]). This method has been applied to study free-energy landscape and folding mechanisms of several peptides and proteins [33], through several variants of the traditional temperature dependent scheme, available in some of the most popular MD packages, such as AMBER [39], GROMACS [40] and NAMD [41].

Among the enhanced sampling methods that fully explore the binding mechanism, metadynamics, especially in the well-tempered formulation, has emerged as a powerful approach for accelerating rare events [35]. This has been applied in drug docking to protein and enzymes, systems involving big conformational changes and relevant solvation effects. Directed dynamics such as the adaptive biasing force (ABF) and hyperdynamics were also derived from the same principles as adopted by metadynamics [35]. The implementation of metadynamics in MD codes, such as NAMD and GROMACS, have promoted a broad range of applications of the method, ranging from solid state physics to biological systems.

The adaptive biasing force algorithm (ABF) has also emerged as a promising strategy for mapping complex free-energy landscapes [42, 43], as it combines both constrained and unconstrained simulations into a highly efficient scheme, providing an uniform sampling of the order parameter. Briefly, as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it produces a Hamiltonian devoid of an average force acting along the transition coordinate of interest. In contrast to umbrella sampling schemes, based on probability distribution functions, ABF uses forces, which can be readily estimated without the need to sample broad ranges of the order parameter.

Another relevant problem in the early estimations of free-energy is related to the strong dependence on system size, in the presence of significant electrostatic interactions [44]. Once long-range corrections using Ewald lattice summation or the reaction field are included in molecular simulations, size effects in neutral systems decrease markedly. The problem, however, persists in charged systems, for example in determining the free energy of charging a neutral species in solution. In this context, it has been demonstrated that system-size dependence can be largely eliminated in these cases by careful treatment of the self-interaction term, which is associated with interactions of charged particles with their periodic images and a uniform neutralizing charge background [44].

The ability to decouple relevant energy contributions from individual components is also far from a simple solution, being the basis of alchemical free-energy methods [19, 20], such as the thermodynamic integration (TI) [17, 44, 45], free-energy perturbation (FEP) [20] and molecular mechanics-Poisson-Boltzmann surface area (MM-PBSA) [46].

conventional MD calculations even when specialized hardware is employed. This well-recognized limitation of MD has led to the development of various algorithms to enhance the sampling of the high free-energy states and rare events, associated to high free-energy barriers.

Enhanced sampling methods can speed up conformational sampling by various means and represent an effective alternative to access with high accuracy the thermodynamics and possibly the kinetics that underlie these processes. Steered MD and umbrella sampling [31], with potential of mean force estimation (PMF) [32] allows inspecting the free-energy profile and

In recent years, other strategies aimed at the same goal have been proposed. Replica exchange [33] metadynamics [34, 35], accelerated MD [36], milestoning [37], transition path sampling [38], and combinations of these are among the most widely used methods to enhance conformational sampling. For instance, the replica exchange methodology has been used in the atomistic simulations, as well as in a number of coarse-grained simulations (see e.g. [34]). This method has been applied to study free-energy landscape and folding mechanisms of several peptides and proteins [33], through several variants of the traditional temperature dependent scheme, available in some of the most popular MD packages, such as AMBER [39],

Among the enhanced sampling methods that fully explore the binding mechanism, metadynamics, especially in the well-tempered formulation, has emerged as a powerful approach for accelerating rare events [35]. This has been applied in drug docking to protein and enzymes, systems involving big conformational changes and relevant solvation effects. Directed dynamics such as the adaptive biasing force (ABF) and hyperdynamics were also derived from the same principles as adopted by metadynamics [35]. The implementation of metadynamics in MD codes, such as NAMD and GROMACS, have promoted a broad range of applications of

The adaptive biasing force algorithm (ABF) has also emerged as a promising strategy for mapping complex free-energy landscapes [42, 43], as it combines both constrained and unconstrained simulations into a highly efficient scheme, providing an uniform sampling of the order parameter. Briefly, as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it produces a Hamiltonian devoid of an average force acting along the transition coordinate of interest. In contrast to umbrella sampling schemes, based on probability distribution functions, ABF uses forces, which can be

Another relevant problem in the early estimations of free-energy is related to the strong dependence on system size, in the presence of significant electrostatic interactions [44]. Once long-range corrections using Ewald lattice summation or the reaction field are included in molecular simulations, size effects in neutral systems decrease markedly. The problem, however, persists in charged systems, for example in determining the free energy of charging a neutral species in solution. In this context, it has been demonstrated that system-size dependence can be largely eliminated in these cases by careful treatment of the self-interaction term, which is associated with interactions of charged particles with their periodic images and a

readily estimated without the need to sample broad ranges of the order parameter.

the mechanistic aspects involved in formation of host-guest complexes [5].

the method, ranging from solid state physics to biological systems.

uniform neutralizing charge background [44].

GROMACS [40] and NAMD [41].

10 Molecular Dynamics

The feasibility of the estimated binding free-energy depends on the correct estimation of thermodynamic quantities and the concerted interaction components (enthalpy, entropy and solvent contributions) [15, 47]. For instance, the quantification of these interaction components have been directed at inclusion complexes between cyclodextrins and its derivatives and different model drugs (see Refs. [13, 15, 48].). However, only a few studies (e.g. [5, 13]) have introduced technical and practical directions to investigate the energy components underlying the stability of these complexes. Spoel and co-workers [13] have emphasized the role of solvent contribution resorting to steered molecular dynamics and PMF calculations. The latter techniques have also been used by Pais and co-workers [5] for assessing the relevance of non-included guest moieties in the binding constants of cyclodextrin inclusion complexes, estimating the weight of enthalpic and entropic contributions to the free-energy. In this context, MD and PMF calculations allow the modulation of the formation processes in inclusion complexes, along which the free-energy profiles are derived from an umbrella sampling procedure [5]. This approach has been employed in the characterization of the inclusion of a wide range of drugs on cyclodextrins, and also in the aggregation of host units either in dimerization [15, 48] or in rotaxanes formation [49, 50]. Other hybrid approaches [51], including quantum mechanics/molecular mechanics (QM/MM) [29] methods have also provided insight on the formation of inclusion complexes at a molecular level.
