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Energy minimization of a molecule
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Objective 

 

 To minimize a molecule so as to obtain a stable conformer.

 

Theory

 

A molecule is a small chemical element that is made up of two or more atoms held together by chemical bonds. These bonds are formed as a result of exchange or sharing of electrons among atoms of the molecule. Atoms have a tendency to attain more number of electrons thereby increasing stability. Bond length is the distance between the nuclei of two bonded atoms whereas bond angle is the angle formed betweeen two adjacent atoms in a molecule. Bond angle and bond length are the two important parameters which determine the shape and size of a molecule. Thus geometry of a molecule can be characterised by analysing the bond length and bond angle. A dihedral angle or torsion angle defines the conformations around rotatable bonds. Dihedral angles of amino acids are biologically important since they define the backbone for protein in protein structure prediction.


Molecular mechanics computes the structure and energy of molecules based on nuclear motion. In this method, electrons are considered explicitly, but assuming that they will find their optimum distribution once the position of the nuclei are known. The assumption is based on Born-Oppenheimer approximation, that nuclei are much heavier than electrons and their approximation is negligibly small compared to the movement of electrons. The steric energy of a molecule can be calculated from the sum of bonded atoms and non bonded atoms.

Molecular mechanics mainly depend upon three parameters - force fields, parameter sets and minimizing algorithms called as triad tools. The potential energy of the molecules in molecular mechanics is calculated by using force field concept.  A force field is a set of functions and constants used to describe the potential energy of the molecule. The parameter set includes atomic mass, Vanderwaal’s radii, bond length, bond angle, dihedral angle which defines reference points and force constants allowing for calculation of potential energy, caused due to the inclusion of attractive or repulsive interactions between atoms. Algorithms to calculate new geometrical positions are so called as minimizers or optimizers. Different methods such as steepest descent, conjugate gradient are available. 

 

The general form of the force field equation is


Epot = ∑Ebon + ∑Eang + ∑Etor + ∑Eoop + ∑Enb+ ∑Eel


where,
Epot is the total steric energy.
Ebon is the energy resulting from changing the bond length from its initial value, which is calculated using Hooke’s equation for the deformation of the spring ( E = ½ kb (b-b0)2  where kb is the force constant for the bond, b0 is the equilibrium bond length and b is the current bond length.
Eang is the energy resulting from deforming a bond angle from its original value .
Etor is the energy resulting from deforming the torsion or dihedral angle.
Eoop is the out-of-plane bending component of the steric energy.
Enb is the energy arising from non-bonded interactions
Eel is the energy arising from coulombic forces.


The main objective of molecular mechanics is to find the lowest energy conformation of a molecule and this process is termed as energy minimization. The lowest energy conformation can be calculated from the bond lengths and angles with smallest steric energy. The system makes several changes in the atom position through rotation and calculates energy in every position. This process is repeated many times to find the position with lowest energy until an overall minimum energy is attained. In every move the energy is kept lowered, otherwise the atom will return to its original position. The one full round of an atom rotation is called minimization step or iteration. By applying force field, the minimum energy of a molecule in its stable conformation can be calculated, which is always taken as negative derivative of the energy function with respect to the coordinates of the atom.


In this study, one would be studying minimization strategy to find the molecules with most stable conformation using Avogadro software. Avogadro is free open source molecular builder software designed for molecular modeling. It allows building of chemical structures, visualization and analysis of molecule, structure optimization, quantum mechanical calculations, electron density calculations, formatting of input and output files etc. Since it has an extensible plug-in based architecture, it helps to perform a wide range of applications. This software is licensed under GNU GPLv2. Avogadro is basically based on C++ language; its latest version uses python scripting. It uses Qt and OpenGL for graphical section and is introduced by Sourceforge service, which is available for Linux, Mac OS X and Windows platforms. It uses OpenBabel to load and save files which is free software that is used to convert chemical file formats. Using Open Babel, different file types like PDB, CML, xyz, etc., can be opened with Avogadro after converting them to suitable file type supported by Avogadro.

 

When the molecule is optimized with its geometry, it normally attains most stable conformation. Different types of force fields can be applied to Avogadro tool with OpenBabel library according to user’s choice of molecules. OpenBabel is free software used to search, convert, analyze, or store data from molecular modeling and other associated areas. MMFF94(s) is mainly used for organic chemistry molecules and drug - like molecules. MMFF94 and MMFF94(s) differ only in the torsion and Out-Of-Plane bending parameters but uses the same function to find its energy. The “s” stands for static, used mainly in parameters where the output is static. UFF can be used across the entire periodic table. For simple organic molecules, Ghemical can be used. Different algorithms such as steepest descent minimization, are used to reduce highly unfavorable clashes. Conjugate gradient minimization on the other hand is much slower but more useful at reaching an energy minimum after severe other clashes.

 

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