Quantum Chemistry and Chemometrics Applied to Conformational Analysis

Molecular structure plays a special role in science. Knowledge of the atomic arrangement is essential in order to be able to elucidate chemical properties and processes. The first advances in determining molecular structure occurred in nineteenth century. Around 1812, Jean-Baptiste Biot, a French physicist, discovered optical activity by observing polarized light shifting when crossing a quartz crystal. He observed that the light was displaced to the right in some cases and to the left in others. The conclusion was that rotation of polarized light by quartz is an inherent property of the crystal. Interested in the phenomenon, Biot noticed in further studies that similar effects were found when polarized light passed through certain liquids such as natural oils (lemon extract and laurel), alcoholic solutions of camphor, some sugars and tartaric acid. (Drayer, 1993; Cintas, 2007; Gal, 2011) Biot’s observations were very important in laying foundation for the concept of optical activity.

105 molecular systems, making it possible to produce three-dimensional representations that provide insights into their behavior.As computer tools have enjoyed a spectacular increase in last decades, theoretical methods are invariably associated with computer modeling.This has become a powerful tool for evaluating molecular structure, from which special chemical information about molecular behavior can be inferred.(Pietropaolo et al., 2011)

Statement of the problem
For the theoretical and computational determination of molecular properties it is necessary to previously determine the minimum energy structure of the system being studied.A central issue is to probe the equilibrium configuration of the molecular system.The way that energy varies with the coordinates is usually referred to as the potential energy surface.At the atomic level, the interaction energy between atoms is essentially ruled by quantum mechanics, which provides the basic elements and methods used in molecular modeling.However, the potential energy surface can be addressed with different degrees of approximation, i.e., ab initio, effective potentials or even more coarse-grained potentials.Irrespective of the details with which the system is considered, one usually faces the problem of a highly dimensional system with the occurrence of multiple minima.Low energy minima play an important role in determining molecular properties, and the determination of these minima conformational states is a non-trivial task, usually referred to as energy minimization method for exploring the energy surface.
If four or more atoms are connected in chain by single bonds we can suppose that there is considerable flexibility in the molecule.The existence of hindered rotation about a single bond is one of the fundamental concepts in conformational analysis.(Mo & Gao, 2007) The understanding of the connections between the atoms is related to the internal coordinate parameters, i.e., bond length, bond angle and dihedral angle, and is essential in designing molecular models.For instance, let us consider a molecule composed of four different atoms which are singlebond linked (Figure 1).The green arrows represent the bond stretch and the average value is the bond length; the red arrows correspond to the angle formed by three sequential atoms, i.e., the angle bond; the curved blue arrow indicates the free rotation around the only single bond able to perform changes in the molecule conformation, as shown in Figure 2: In other words, different conformations are obtained when a dihedral angle is rotated.A dihedral angle is that composed by the planes formed by the sequence of three atoms (Figure 3): The main questions on molecular modeling are concerned with a good way of finding the global minimum energy structure.Important information is also concerned with the behavior of the conformational space.It is not only necessary to know which is the global minimum, but also the whole shape of the potential energy surface (PES).The main characteristic of this conformational phase space is that it is exponentially large, and a computationally hard problem.(Fraenkel, 1993) One problem that illustrates this difficulty is the protein folding, in which one searches for the global energy minimum structure associated with its functional conformation.If a method can be used to describe the relevant potential energy surface of a given molecule, it can also accurately elucidate its behavior against many interest situations.
Many techniques have been presented, and it is not the goal of this work to make a deep study on them.A good discussion of practical methods is given by Leach.(Leach, 2001) We intend to give a brief idea of the most popular techniques used for investigating the molecular structure computationally.For conformational sampling, one can imagine a hierarchy of methods with different computational costs.(Seabra et al., 2009) However, there is no sovereign truth about what is the best method for performing a conformational analysis.Each situation must be evaluated.The best method is one that has the best fit to the problem studied; in practical terms it will provide the answer as quickly as possible, using the least amount of computer resources.

Stochastic methods of conformational analysis
The literature reports many methods for trying to solve multiconformational problems, and most of them are based on stochastic approaches.Put simply, stochastic methods work with random variables, such as initial conformations for the search or the steps probing the configuration phase space.The simple criterion for establishing a minimum energy conformation is that the first derivatives of the energy E with respect to each variable (x i ) is zero and the second derivatives are all positive: The algorithms that search for minimum energy states can be classified into two groups: those which use derivatives of the energy with respect to the coordinates, and those which do not.The most used derivative minimization methods are the steepest descent, line search in one dimension and conjugate gradient methods.(Leach, 2001) These algorithms are very useful for conducting local (restricted) searches of minima, or downhill searches to the nearest minimum, since they are not able to overcome energy barriers.They are often used in combination with other stochastic methods.
In the remainder of this section we discuss examples of stochastic methods.Havel, Kuntz and Crippen described distance geometry algorithms in conformational analysis.(Havel et al., 1983b) Given the impossibility of examining all possible conformations, they introduced a method which is capable of finding global optima without considering all possible solutions by means of combinatorial optimization.The method is known as brunch and bound and involves logical tests that allow whole classes of solutions to be eliminated without examining them one by one.The method converts a set of distance ranges (or "bounds") into a set of Cartesian coordinates that are consistent with those bounds.(Spellmeyer et al., 1997) The efficiency of a branch and bound algorithm depends on how effective these tests are compared to the time required to perform them.(Havel et al., 1983a;Havel et al., 1983b) In another study, Havel et al presented the basic theorems of distance geometry in Euclidean space.They proposed new algorithms and described refinements to the existing ones.All these algorithms were similar because they utilize geometric principles in order to interpret structural relationships.(Havel et al., 1983b) According to Leach and Smellie,(Leach & Smellie, 1992) distance geometry is a method for searching conformational space in which a structure is initially formulated in terms of interatomic distances.Any molecular system can be described as the set of minimum and maximum interatomic distances between all pairs of atoms in the molecule.The complete conformational space of the molecule is contained within this space.In distance geometry, a matrix is defined as the set of minimum and maximum distances, and then used to create a series of conformers that are consistent with those distances.(Spellmeyer et al., 1997) Another tool for performing conformational searches is the genetic algorithm, a stochastic method first introduced by Holland in 1975.Genetic algorithm (GA) is a method applied to solve problems using a natural evolution process simulation.It is a stochastic method developed in analogy to Darwin's theory of evolution in order to perform the optimization.(Brodmeier & Pretsch, 1994;Lucasius, 1993;Nair & Goodman, 1998) Genetic algorithm is commonly used for studying a large-scale space of possible solutions.The goal is to identify the best solutions within that space without the need to evaluate all possibilities.(Yanmaz et al., 2011) The GA is the optimization of a large number of possible solutions using a randomly generated population.When applied to conformational analysis, the population of interest consists of different conformations.The biological evolution of this population is simulated.A population of trial solutions is iteratively manipulated by a series of genetic operators to satisfy an objective function.The adjustment is calculated, and a new population is generated according to operators, such as selective reproduction, recombination and mutation.The process is repeated until the minimum energy structures are obtained.(Lucasius & Kateman, 1994;Beckers et al., 1996;Beckers et al., 1997) Artificial Neural Networks (ANN) are another example of stochastic methods used in conformational analysis.This method is based on concepts of the behavior of the human brain.Although artificial neural networks are primitive compared to their biological counterparts, they exhibit some interesting properties which make them useful as multivariate tools in various fields of research.During the last decade, ANN have been successfully applied in non-linear modeling, classification, signal processing and process control.(Derks & Buydens, 1996) The properties of a molecule are intimately linked to the conformations that it adopts and so an understanding of the conformational space is important in rationalizing and predicting its behavior.(Jordan et al., 1995) Among the most popular stochastic methods for covering the conformational space are Monte Carlo (MC) and Molecular Dynamics (MD).They are similar in the sense that both procedures include the same representation of molecules and use classical force fields for the potential energy terms, under periodic boundary conditions.The main purpose of these methods is to sample the phase space and to use the force fields ability to represent the conformational space www.intechopen.comnear minima and connecting transition structures.(Jorgensen & TiradoRives, 1996;Grouleff & Jensen, 2011) However, large differences are found in sampling and configuring space available to the system.For MC, a new configuration is generated by selecting a random molecule or part of it, rotating it, translating it, and performing an internal structural variation.These changes do not necessarily need to follow a realistic physical trajectory.The acceptance of the new configuration is, however, determined by the Metropolis sampling algorithm.The sampling criterion is set in a way that enhances the likelihood of probing low energy conformations.Application over enough configurations yields properly Boltzmann-weighted averages for structure and thermodynamic properties.For MD, given a set of initial conditions (position and velocities of all atoms), new configurations are generated by application of Newton's equations of motion, so that the new atomic positions and velocities of all atoms are determined simultaneously over a small time step.In both cases, the force field controls the total energy (MC) and forces (MD), which determines the evolution of the systems.(Jorgensen & TiradoRives, 1996) Examples of problems related to large systems are the interaction between drug and the receptor, and protein behavior and folding.Molecular docking procedures are capable of predicting the three-dimensional structure of macromolecular complexes and their binding affinity.The information required is simple and corresponds to the structures of the receptor and ligand and the presumable interfacing region between them.Besides the simplicity of these docking procedures, they have low computational costs.However, molecular plasticity and solvation effects are not, or are only approximately, taken into account in these approaches.Free energy simulations may be then used to investigate the molecular association process and to predict binding affinity.(Biarnés et al., 2011) It is important to realize that sometimes the probing of PES addresses singular questions, which involve association of several methods, also called hybrid methods.A particular wellknown tailored one is the quantum mechanics/molecular dynamics approach, also known as QM/MM approach.This is a molecular simulation method that combines the strength of both QM (high accuracy in specific regions) and MD fast calculations (in not so crucial regions), in such a way that it efficiently allows the study of chemical processes in solution and in proteins.
When stochastic methods are used to find minimum energy conformations, asymptotic states in restricted regions of the phase space are probed.This means that there is no end point in the search, and the convergence cannot be assured.

Systematic search in conformational analysis
As seen before, stochastic techniques use different heuristics to randomly cover the conformational space.These algorithms apply a perturbation to the initial conformer and minimum energy conformation is associated with the lowest energy state that is found through out this procedure.They provide a sampling of energy minima structures and the shape of the PES is obtained in an indirect way.
Beyond the stochastic methods there are procedures that do not work with random choice to cover conformational space.These classes of methods are described as deterministic and are capable of searching the conformational map in a systematic way, providing a direct www.intechopen.comQuantum Chemistry -Molecules for Innovations 110 knowledge of PES shape.These searches divide conformational space into quantized units and apply algorithms to search this discrete space or define a set of heuristic rules that are used to drive the search.(Smellie et al., 2003) Systematic methods are those that explore all conformational space at some fixed degree of resolution.To perform the systematic search, a molecule must be numerically described by its atoms' internal coordinates.The internal coordinates are bond length, angle bond and dihedral (torsion) angle.For a given initial structure the systematic conformational search is conducted by regular variation in dihedral angles (Figure 2).
Although a systematic search can obtain the morphology of a molecule's energetic behavior directly, this method is not feasible for evaluating complex systems.(Beusen et al., 1996) Systematic search is most usefully applied for molecules with few degrees of freedom.(Li Manni et al., 2009) According to literature (Beusen et al., 1996), to cover the PES corresponding to the conformational space, different molecular structures must be systematically generated by rotating the torsion angles around the single bonds between 0° and 360°.The number of conformations is given by: where N is the number of free rotation angles, and s is the number of defining steps according to the angle increment: with θi being the dihedral increment of angle i.
An examination of equation ( 1) reveals that the number of conformations generated will exponentially increase in proportion to the number of bonds with free rotation in the molecule under study.A problem arises if the number of steps is large, i.e., when a very refined surface is required by small angle increments.This problematic behavior of the systematic study of PES, described as combinatorial explosion, is the major restriction involved in this kind of search.Figures 4 and 5 illustrate how combinatorial explosion works.In Figure 4, we have a representation of the system growth where many single bonds can be rotated.
The combinatorial explosion problem is represented by Figure 5.The number of branches to be considered is shown by the ramification achieved according the number of angles (A, B, C, D…) and will depend on the dihedral increment chosen.Due to the problem involved in combinatorial explosion, systematic search becomes nonviable for studying large molecules, since the number of degrees of freedom increases.A useful strategy for reducing the dimensionality of the conformational space is to perform systematic conformational searches on small portions of the molecule (either as isolated fragments or in situ).Using these optimal parts, one builds the conformation of the whole molecule with only limited additional searching of the relative conformations of the fragments.Approaches that incorporate this principle are known as ''build-up'' methods.(Beusen et al., 1996;Izgorodina et al., 2007) There are some strategies for overcoming the combinatorial explosion.We will focus our discussion on procedures that involve chemometrical approaches.

Chemometrics and structure determination
A conformational search, independently of the method chosen, usually involves large amounts of data.Sometimes, data achieved from a given methodology must be explored by an additional technique.According to Geladi (2003) "data exploration means taking a look at the data to find interesting phenomena, often without prior expectations.As a result, outliers, clustering of objects and gradients between clusters may be detected."(Geladi, 2003) Chemometrics has been used extensively in recent years for exploring chemical problems by means of computer tools and statistical observations.The literature presents many definitions for chemometrics.For our purposes, this field of knowledge is better defined as a combination of two definitions found in the literature: a.According to Wold (1995), chemometrics can be understood as a way "to get chemically relevant information out of measured chemical data, how to represent and display this information, and how to get such information into data"; (Wold, 1995) b. For Beeb (1998), chemometrics corresponds to "the entire process whereby data (e.g., numbers in a table) are transformed into information used for decision making."(Beebe, 1998) The above definitions indicate that chemometrics offers a broad approach to chemical measurement sciences.It is not restricted to the actual experimental analysis but also considers what happens before and after it.(Massart et al., 2004) It is the goal of chemometrics to extract the information from the data.(Ramos et al., 1986) Chemometrical approaches have been applied to conformational analysis for handling special difficulties of large amounts of data generated both by stochastic and by systematic searches.
Among the various chemometrical techniques, Principal Component Analysis (PCA) is the most commonly used for conformational problems.In many ways, it forms the basis for multivariate data analysis.PCA is a multivariate method of analysis whose main concern is to reduce the dimensions needed to portray accurately the characteristics of a large dimensional data matrix.(Beebe, 1998;Wold et al., 1987) This mathematical procedure consists of eliminating a large number of correlated variables without changing the characteristics of the original data-set that contribute most to its variance.
For an easy graphical representation, consider a two-dimensional set of variables as shown in Figure 6 (a).
PCA can be performed on the original variables as shown in Figure 6 (b) and new axes, called Principal Components, arise to account for the maximum variation.A subsequent rotation (Figure 6(c)) is made on these new PC axes in order to rewrite the original variables in terms of this new axes-system.Each PC is constructed as a linear combination of variables: where Pi is the ith principal component and ci,j is the coefficient of the variable xi,j.(Leach, 2001) There are ν such variables.The first principal component PC1 is chosen in order to maximize the data variance of the axis.The second and subsequent ones are chosen to be orthogonal to each other and account for the maximum variance in the data not yet described by previous principal components.A variety of algorithms can be used to calculate the principal components.The most commonly employed approach is singular value decomposition SVD.(Golub & Loan, 1996) Fig. 6.PCA procedure: (a) original data set; (b) PCA on original data set and (c) Variables according to the new PC coordinates.(Beebe, 1998) A matrix of arbitrary size can be decomposed into the product of three matrices in such a way that: where U and V are square orthogonal matrices.The matrix U (whose columns are the eigenvectors of XX t ) contains the coordinates of samples along the PC axes.The V matrix (which contains the eigenvectors of the correlation matrix X t X) contains the information about how the original variables were used to make the new axes[ci,j coefficients in eq. ( 3)].The S matrix is a diagonal matrix that contains the eigenvalues of the correlation matrix (standard deviations) or singular values of each of the new PCs.The diagonalization of symmetric matrices (such as XX t and X t X) and SVD are fundamental problems in linear algebra (Golub & Loan, 1996), for which computationally efficient software has been developed and can be used on a routine basis (Hanselma et al., 1997) for very large-size matrices.
In chemistry, PCA was introduced by Malinowski around 1960 under the name Principal Factor Analysis, and further developed after 1970.(Malinowski, 2003) Principal Component Analysis can be used for crystallographic structure data; in its general form, conformational analysis is applied to multivariate numerical problems.(Allen et al., 2010) Many studies report on the use of PCA for handling Molecular Dynamics data.Among them, we highlight the application that uses PCA for mapping potential energy surfaces, by the quantitative visualization of a macromolecular energy funnel.(Becker, 1998) Other examples where PCA can be applied in molecular structure determination can also be found in recent studies.(Das et al., 2011;Araujo-Andrade et al., 2010;Kiralj et al. 2007;Oblinsky et al., 2009;Silva et al., 2011)

Pairs of dihedral angles-systematic analysis
There is a variety of theoretical methods that are capable of locating minimum energy structures in the potential energy surface.The problem of stochastic methods is that there is no natural end point for the conformational search.In some cases, only a small subset of conformational space is explored and the convergence of the system is not guaranteed.Only Systematic Conformational Analysis maps the conformational space completely.We stress the principal difficulty inherent in this method is the combinatorial explosion.In a previous study (Bruni et al., 2002), a new methodology was introduced for controlling the combinatorial explosion through a systematic reduction in the size of the system by means of chemometrics.
This method consists of a small systematic conformational analysis, in which the conformational space is studied by rotation of the important free rotation in pairs, described as Pairs of Dihedral Angles-Systematic Analysis -PDA-SA.The main objective is to reduce the dimension of the investigated system.The idea is to address the conformational space in small portions, evaluating PES in combinations of angles in pairs.If the problem of combinatorial explosion is controlled, the conformational space can be sufficiently refined in the regions of minimum energy, taking care to minimize the information lost.The energy surfaces are obtained for each pair of angles and the number of conformations is given by Equation 5: where s and N have the same meaning as in Equation 1.The number of conformations, in this case, is given by the combinatory arrangement of the N dihedrals in pairs.
The main observation of the comparison between equations (1) and ( 5) is that the number of conformations as given by Eq. ( 1) increases exponentially with the number of bonds with free rotation, while from Eq. ( 5), the number of studied conformations increases quadratically with N. As the number of free rotation angles increases, the difference in the number of conformers generated by these two equations becomes more evident.
The computational procedure for PDA-SA can be organized in five basic steps: 1. Molecular Building: The interest molecule must be defined in terms of its internal coordinates: bond length, angle bond and dihedral angles.There are many softwares able to define this molecular initial structure.A quantum chemistry optimization is required at this step in order to adjust internal parameters.The best method must be chosen according to the system under study.

Dihedral Pair Rotation:
The PDA-SA conformational search begins and the combination of the existing pairs of angles is taking account.Sometimes it is only possible to choose a dihedral increment with a less refined value.A rough PES is obtained in this case.The matrix to be analyzed consists of energy values from potential surfaces for angle combinations, and they are grouped according to Figure 7 for N angles.Appendix A shows the energy values for omprazole basic structure.The idea is to perform a cyclical permutation on the data, and this matrix form ensures that no information about the total PES is lost.The energy values obtained for each angle rotation as a function of the others allow the conformational space to be completely mapped.The major advantage is that the shape of these small portions can be visually observed, since we have a 3-D fitting.(see Figures 9 and 10) 3. PCA application on data matrix: After the energy matrix statement, PCA is performed on the data.The regions with minimum energy points on the grid search can be easily selected.The number of selected regions will depend on the nature of the studied system.4. Refinement with a short dihedral increment: The regions initially obtained in step 3 can be refined with a small angle increment.It is important to emphasize that this step is not obligatory, since a small dihedral increment can be used in step 3, depending on the studied system.However, previous experience in this methodology (Bruni et al., 2002;Bruni & Ferreira, 2008) shows that this is the easiest procedure, i.e., firstly make rotations with a large dihedral increment and subsequently refine the minimum energy regions selected by PCA with small dihedral increments. 5. Optimization of the final structure: the procedure described above provides angle values for the conformational search with a good level of accuracy.When these values are combined, we obtain all the possible minimum structures.Those structures constrained by the angle values obtained by PCA analysis are submitted to final optimization and the resulting structures are considered to be those of minimum energy.
In the study that introduced this method, the approach was successfully tested in the analysis of omeprazole and its derivatives, in which the results were in agreement with the experimental ones.(Bruni et al., 2002) In a second study, the technique was used to find minimum energy conformations of omeprazole derivative molecules in a QSAR study.(Bruni & Ferreira, 2002) It was shown that conformational analysis is crucial when establishing SAR/QSAR models using theoretically calculated descriptors, and they are strongly dependent on the details of molecular structure.Though all minima conformation have similar energetic values, some calculated properties are very sensitive to the structural variation, which is understandable since electronic properties are intrinsically dependent on molecular conformation.(Bruni & Ferreira, 2002) Omeprazole's racemization barrier and decomposition reaction was also studied.Quantum chemistry coupled to PDA-SA chemometric method was used to find all omeprazole minimum energy structures.To obtain the racemization barriers it was essential that the starting structure was in a global energy minimum.In that work, for all the studied structures, there was no change in the values of the racemization barriers, which confirmed the identification of the most stable structures for omeprazole.(Bruni & Ferreira, 2008) Fig. 7. Matrix scheme for N angles: the discrete energy values for each rotation angle must be evaluated.E ij are the energy matrices with elements E ij km , in which k and m are the angle increment indices for the angles i and j, respectively.
This approach is straightforward and in principle would have no size limits for its application.However, it presents limitation due to some initial condition dependence.
Given a system with N degrees of freedom, for each pair of angles there are N-2 parameters that can interfere in the method.For example, in Figure 4 the potential energy surface for first and last dihedral angle combinations depends on the dihedral angles conformation between them.When the dihedral angles are too far from each other along the chain of atoms, the method may not become feasible.In this case the method may need to be repeated with different initial conditions to improve the sampling of the configurational phase space, and moreover we cannot be sure that we have reached the global minimum.
When the correlations between the pairs of angles do not depend strongly on these initial conditions the method is very useful.Such system corresponds to small molecules, not so flexible, in which there are few large potential basins, such as omeprazol and its derivatives.(Bruni et al., 2002;Bruni & Ferreira, 2002;Bruni & Ferreira, 2008) The limit of validity for this method is under investigation.We are applying this method to study the IAN peptide, which is a tetrapeptide isobutyryl-(ala)3-NH-methyl.(Nascimento et al., 2009) This is the smallest polypeptide that can have secondary-like structure (an helix) (Becker & Karplus, 1997) and it has 11 free rotation bonds.For a flexible system, such as this, the initial condition dependence in the calculation of the minimum energy conformations is expected to increase with the size of the system.Since the system is more flexible and expected to be more rugged, we partially overcome this problem by using small angle increments steps, in order to probe all local minima of the system and compare them.

Study of basic structure for omeprazole and derivatives
Initially, the basic structure of omeprazole and derivatives was evaluated.This structure has three bonds with free rotation.To validate the proposed methodology, two different approaches were performed.In the first approach, pairs of angles were taken account ((1,2), (1,3) e (2,3) in Fig. 8) and the number of conformations is given according to Equation 5.The resulting matrix analyzed was composed by the energy values from the potential energy surface for each angle combination (see matrix example in Figure 7).A matrix with discrete energy values for the basic structures with 30° angle increment in Equation 2 is showed in Appendix A. Three PES were obtained for a 30° dihedral increment and are showed in Figures 9 and 10.
Figure 9 shows the original energy values and Figure 10 shows the same surfaces, but with a 0,12 hartrees cut off for better visualization.PCA was performed on autoscaled original data and the results are shown in Figure 11.64% of the whole information is cumulated in first and second Factors (or Principal Components-PCs).The convergence of the points for one region is observed.Figure 12 shows the PCA results for the leveled data in 0,12 hartrees.Factor 1 and Factor 2 now cumulate 73% of the entire information.Factor 1 accounts to the minimum region in each case and Factor 2 accounts for the energy range for the different combinations.Table 1 shows the selected minimum energy for each angle.The first column shows that two different regions were chosen for Angle 1 and only one region for Angles 2 and 3. Second column shows the rotation over the initial angle value (third column) resulting in the fourth column.Once minima energy regions were defined, a small angle increment (5°) was used on them.
Results for PCA are in Figure 13.In all cases a parabolic behavior was observed.When data variation decreases, curves are more easily observed and the minimum point is detectable.
The amount of information accounted for both first and second PC´s (Factors) is around 90%. Table 2 shows the final values for each angle.When these values are combined, two different geometries were obtained with similar energy values (Table 3).These conformations are shown in Figure 14.In the second approach, the conformational analysis was made according to Equation 1 and took into account all possible conformations.PCA was performed on data matrix and minimum energy regions were selected.The next step a lower dihedral increment of 5° was used to refine those selected regions.PCA was performed again, and the same structures and energy, shown in Table 3, were obtained.This indicates that the two approaches are equivalent.The details of this complete systematic search can be found in (Bruni et al., 2002).

Number Energy
The second group is composed by final conformations 4, 7 and 8 (green ones in Table 5).The superposed conformations can be observed in Figure 18.These conformations are more open and have only one hydrogen-bond (red circle in Fig. 18).The last group, the black ones in Table 5, are superposed in Figure 19.The resulting structures show an alpha-helix like behavior, with two stabilizing hydrogen bond (red circles, Fig. 19).
Results presented for IAN peptide are partial and were only performed for one minimum region for each starting structure.Other minimum energy regions of this system are being investigated.A gradual increase in the size of the chain is also been explored.

Conclusion
The arrangement of atoms in a molecule or its structure determination has intrigued scientists through history.However only with recent experimental and computational advances the discussions on this theme became more effective and elucidative.The nature of PES is intrinsically multidimensional, usually has a very complex landscape.The global minima search, like the one encountered in the protein folding problem, is a NP-hard problem.This means that this task belongs to a large set of computational problems, assumed to be very hard ("conditionally intractable") (Fraenkel, 1993).The search for its relevant minima in molecular modeling has motivated the development of methods with very specific applications, as discussed in this chapter.For each particular problem one finds a variety of methods that allows feasible solutions, and most likely a combination of methods provides the optimum solution.
In this chapter, we discussed some aspects of conformational search that controls the combinatorial explosion.In particular, Principal Component Analysis was associated with a systematic search method to find structures with low energy in PES.The methodology can be useful to handle small-and medium-size molecules.The maximum size which the method can efficiently handle is being investigated (Nascimento et al., 2009).Due to the PCA dimension reduction, the method's efficiency is highly increased, allowing it to be of practical use in the study of more complex molecules.

Apendix A
Matrix with the discrete values for each rotation angle and its corresponding energy value for the first rotation for basic structure in Figure 8. Labels in bold were not used in PCA analysis, they are shown to help the matrix notation and visualization.

Fig. 1 .
Fig. 1.Model for a generic molecule with four different atoms.

Fig. 4 .
Fig. 4. A general structure with many single bonds.

Fig. 5 .
Fig. 5. Branches generated by the dihedral angles A, B, C and D.
Figure15shows the IAN 2D structure www.intechopen.comQuantum Chemistry -Molecules for Innovations 122(Becker, 1998).Red arrows indicate the , Φ e dihedrals.The dihedral angles , and Φ are related to the rotations of single bonds between atoms in the main chain C (i)-C, OC-NH and N-C(i+1), respectively, where C (i) is the ith alpha carbon of the polypeptide chain.Angles and Φ are connected to two arrays of functional protein chain: alpha-helix or beta-sheet.