**1. Introduction**

Molecular modeling relies on the event of theoretical and computational methodologies, to model and study the behavior of molecules, from little chemical systems to big biological molecules and material assemblies. The applying fields of molecular modeling regard computational chemistry, drug design, computational biology and materials science. The fundamental computational technique to perform molecular modeling is simulation. Molecular simulation techniques need specific extra computational and code software system [1]. Most molecular modeling studies involve three stages. Within the initial stage a model is chosen to explain the intra- and inter-molecular interaction within the system. The two most typical models that are utilized in molecular modeling are quantum mechanics and molecular mechanics. These models enable the energy of any arrangement of the atoms and molecules within the system to be calculated and permit the modeler to work out how the energy of the system varies because the positions of the atoms and molecules change. The second stage of a molecular modeling study is that the

calculation itself, such as an energy Minimization, a molecular dynamics or Monte Carlo simulation, or a Conformations search. Finally, the calculation should be analyzed, not solely to calculate properties however additionally to see that it's been performed properly [2].

In molecular modeling we tend to are particularly curious about minimum points on the energy surface. Minimum energy arrangements of the atoms correspond to stable states of the system; any movement off from a minimum provides a configuration with a better energy. There is also a really sizable amount of minima on the energy surface. The minimum with the very lowest energy is known as the global energy minimum. To spot those geometries of the system that correspond to minimum points on the energy surface we tend to use a Minimization algorithm. The highest point on the pathway between two minima is of particular interest and is understood as the saddle point, with the arrangement of the atoms being the transition structure. Both minima and saddle points are stationary points on the energy surface, wherever the primary derivative of the energy function is zero with regard to all the coordinates [3].

A geographical analogy is useful thanks to illustrate several of the ideas as during this analogy minimum points correspond to the lowest of valleys. A minimum is also represented as being in an exceedingly 'long and slender valley' or 'a flat and featureless plain'. Saddle points correspond to mountain passes. Confer with consult with algorithms creating steps as 'uphill' and downhill'.

#### **1.1 Energy minimization: a brief description about the problem**

The Minimization problem can be formally stated as follows: given a function *f* which depends on one or more independent variables *x*1*, x*2*,….., x*i*,* find the values of those variables where *f* has a minimum value. At a minimum point the first derivative of the function with respect to each of variables is zero and the second derivative are all positive:

$$\left\| \begin{array}{c} \widehat{\boldsymbol{\mathcal{O}}} \;/\,\partial \mathfrak{x}\_{i} = \mathbf{0}; \; \widehat{\boldsymbol{\mathcal{O}}}^{2} \; f \;/\,\partial \mathfrak{x}\_{i}^{2} > \mathbf{0} \end{array} \right. \tag{1}$$

**121**

**Figure 1.**

*the closest energy minimum.*

*Energy Minimization*

software packages [4].

*DOI: http://dx.doi.org/10.5772/intechopen.94809*

method is used because we do not have any Minimization method developed which could be applied to all. Any method which is developed for efficient performance with quantum mechanics may or may not be compatible for molecular mechanics because quantum mechanics deals with models having very less atoms as compare to molecular mechanics. Another point is that procedures like inversion of matrix in some Minimization methods works fine for small systems but problem arises when number of atoms increases to thousands. To calculate the number of derivatives of different Conformations and their energies, different level of performance is required for quantum mechanics than molecular mechanics. Molecular mechanics requires an algorithm that is having more number of steps; quantum mechanics has the opposite scenario. Therefore we have various methods in various popular

As, most of the Minimization algorithms can only identify the minimum energy point which is closest to the starting point, thus it can be stated that they only move downwards or more appropriately downhill on the energy surface. Suppose, this schematic energy surface is shown in **Figure 1**, having three starting points A, B and C to obtained the minima. The locations at which any hypothetical ball stops rolling on energy surface under gravitational force will have corresponding energy minima. But more important thing is to identify global energy minimum which can only be generated by using different starting points, which will be minimized later. Using this criterion, some of the Minimization methods can move uphill to find out energy minimum than the closest one. But not a single algorithm till date is reported for efficiently identification of the global minimum energy from a random starting point. To identify the number of different minimum energy Conformations the shape of energy surface is very useful. For example, population or number of minimum in a deep and narrow valley will be very less than population at broad minimum because it is having higher energy as the vibrational energy is more widely spaced in the minimum and so less accessible. Therefore, the global energy minimum may not be the most highly populated minimum. Thus, there may be the case that the 'functional' structure (e.g. the biologically active conformation of a drug molecule) may not belong to the global minimum, or to the most highly populated conformation, or even to a minimum energy structure at all [3].

Every Minimization algorithm has a set of initial coordinates as input. These coordinates can be generated from different sources. These can be generated using traditional experimental method like X-ray crystallography or NMR. Alternatively, these can be generated by employing a theoretical method like conformational search procedure. But for practical efficiency, both types of methods can be used combinatorially. For example, the behavior of a protein in water can be studied

*A one-dimensional energy surface showing minimization methods movement downwards or downhill towards* 

With respect to present discussion, the most important functions are the quantum mechanics or molecular mechanics energy with the variables *x*i being the Cartesian or the internal co-ordinates of the atoms. It is a common practice to always perform Molecular mechanics Minimizations in Cartesian co-ordinates, in which the energy is a function of *3 N* variables; on the other hand, for quantum mechanics internal co-ordinates are often used. The least value of any function can be identified using standard calculus methods for analytical functions. But, due to the complexities of pattern of energy change with change in the coordinates, it is almost impossible for any molecular system. Therefore, the energy minima are often identified with the help of numerical methods. These methods gradually make changes to the coordinates to generate configurations having lower and lower energies until the minimum is reached [2].

Minimization algorithms can be classified into two categories: one in which we use derivatives of the energy with respect to the coordinates and second in which we do not use any derivative. Derivatives are extremely important because they have details about the shape of the energy surface and due to this efficiency to locate the minimum energy is increases drastically. For any problem, before choosing best algorithm (or algorithms), several points should be considered for e.g. the best Minimization algorithm should use least memory to generate the answer as quickly as possible. For different problems of molecular modeling, different Minimization

#### *Energy Minimization DOI: http://dx.doi.org/10.5772/intechopen.94809*

*Homology Molecular Modeling - Perspectives and Applications*

with algorithms creating steps as 'uphill' and downhill'.

**1.1 Energy minimization: a brief description about the problem**

performed properly [2].

to all the coordinates [3].

derivative are all positive:

energies until the minimum is reached [2].

calculation itself, such as an energy Minimization, a molecular dynamics or Monte Carlo simulation, or a Conformations search. Finally, the calculation should be analyzed, not solely to calculate properties however additionally to see that it's been

In molecular modeling we tend to are particularly curious about minimum points on the energy surface. Minimum energy arrangements of the atoms correspond to stable states of the system; any movement off from a minimum provides a configuration with a better energy. There is also a really sizable amount of minima on the energy surface. The minimum with the very lowest energy is known as the global energy minimum. To spot those geometries of the system that correspond to minimum points on the energy surface we tend to use a Minimization algorithm. The highest point on the pathway between two minima is of particular interest and is understood as the saddle point, with the arrangement of the atoms being the transition structure. Both minima and saddle points are stationary points on the energy surface, wherever the primary derivative of the energy function is zero with regard

A geographical analogy is useful thanks to illustrate several of the ideas as during this analogy minimum points correspond to the lowest of valleys. A minimum is also represented as being in an exceedingly 'long and slender valley' or 'a flat and featureless plain'. Saddle points correspond to mountain passes. Confer with consult

The Minimization problem can be formally stated as follows: given a function *f* which depends on one or more independent variables *x*1*, x*2*,….., x*i*,* find the values of those variables where *f* has a minimum value. At a minimum point the first derivative of the function with respect to each of variables is zero and the second

With respect to present discussion, the most important functions are the quantum mechanics or molecular mechanics energy with the variables *x*i being the Cartesian or the internal co-ordinates of the atoms. It is a common practice to always perform Molecular mechanics Minimizations in Cartesian co-ordinates, in which the energy is a function of *3 N* variables; on the other hand, for quantum mechanics internal co-ordinates are often used. The least value of any function can be identified using standard calculus methods for analytical functions. But, due to the complexities of pattern of energy change with change in the coordinates, it is almost impossible for any molecular system. Therefore, the energy minima are often identified with the help of numerical methods. These methods gradually make changes to the coordinates to generate configurations having lower and lower

Minimization algorithms can be classified into two categories: one in which we use derivatives of the energy with respect to the coordinates and second in which we do not use any derivative. Derivatives are extremely important because they have details about the shape of the energy surface and due to this efficiency to locate the minimum energy is increases drastically. For any problem, before choosing best algorithm (or algorithms), several points should be considered for e.g. the best Minimization algorithm should use least memory to generate the answer as quickly as possible. For different problems of molecular modeling, different Minimization

/ 0; *<sup>i</sup>* ∂∂= *f x* 2 2 / 0 *<sup>i</sup>* ∂ ∂> *f x* (1)

**120**

method is used because we do not have any Minimization method developed which could be applied to all. Any method which is developed for efficient performance with quantum mechanics may or may not be compatible for molecular mechanics because quantum mechanics deals with models having very less atoms as compare to molecular mechanics. Another point is that procedures like inversion of matrix in some Minimization methods works fine for small systems but problem arises when number of atoms increases to thousands. To calculate the number of derivatives of different Conformations and their energies, different level of performance is required for quantum mechanics than molecular mechanics. Molecular mechanics requires an algorithm that is having more number of steps; quantum mechanics has the opposite scenario. Therefore we have various methods in various popular software packages [4].

As, most of the Minimization algorithms can only identify the minimum energy point which is closest to the starting point, thus it can be stated that they only move downwards or more appropriately downhill on the energy surface. Suppose, this schematic energy surface is shown in **Figure 1**, having three starting points A, B and C to obtained the minima. The locations at which any hypothetical ball stops rolling on energy surface under gravitational force will have corresponding energy minima. But more important thing is to identify global energy minimum which can only be generated by using different starting points, which will be minimized later. Using this criterion, some of the Minimization methods can move uphill to find out energy minimum than the closest one. But not a single algorithm till date is reported for efficiently identification of the global minimum energy from a random starting point. To identify the number of different minimum energy Conformations the shape of energy surface is very useful. For example, population or number of minimum in a deep and narrow valley will be very less than population at broad minimum because it is having higher energy as the vibrational energy is more widely spaced in the minimum and so less accessible. Therefore, the global energy minimum may not be the most highly populated minimum. Thus, there may be the case that the 'functional' structure (e.g. the biologically active conformation of a drug molecule) may not belong to the global minimum, or to the most highly populated conformation, or even to a minimum energy structure at all [3].

Every Minimization algorithm has a set of initial coordinates as input. These coordinates can be generated from different sources. These can be generated using traditional experimental method like X-ray crystallography or NMR. Alternatively, these can be generated by employing a theoretical method like conformational search procedure. But for practical efficiency, both types of methods can be used combinatorially. For example, the behavior of a protein in water can be studied

#### **Figure 1.**

*A one-dimensional energy surface showing minimization methods movement downwards or downhill towards the closest energy minimum.*

using its x-ray generated structure. Then place this in a solvent completely. Monte Carlo or molecular dynamics simulation can generate the atomics or Cartesian coordinates of the solvent molecules.
