Background
The "mechanical" molecular model was developed out of a need to describe molecular structures and properties in as practical a manner as possible. The range of applicability of molecular mechanics includes:
‧ Molecules containing thousands of atoms.
‧ Organics, oligonucleotides, peptides, and saccharides (metallo-organics and inorganics in some cases).
‧ Vacuum, implicit, or explicit solvent environments.
‧ Ground state only.
‧ Thermodynamic and kinetic (via molecular dynamics) properties.
The great computational speed of molecular mechanics allows for its use in procedures such as molecular dynamics, conformational energy searching, and docking, that require large numbers of energy evaluations.
Molecular mechanics methods are based on the following principles:
‧ Nuclei and electrons are lumped into atom-like particles.
‧ Atom-like particles are spherical (radii obtained from measurements or theory) and have a net charge (obtained from theory).
‧ Interactions are based on springs and classical potentials.
‧ Interactions must be preassigned to specific sets of atoms.
‧ Interactions determine the spatial distribution of atom-like particles and their energies.
Note how these principles differ from those of quantum mechanics.
The Anatomy of a Molecular Mechanics Force-Field
The mechanical molecular model considers atoms as spheres and bonds as springs. The mathematics of spring deformation can be used to describe the ability of bonds to stretch, bend, and twist:
Non-bonded atoms (greater than two bonds apart) interact through van der Waals attraction, steric repulsion, and electrostatic attraction/repulsion. These properties are easiest to describe mathematically when atoms are considered as spheres of characteristic radii.
The object of molecular mechanics is to predict the energy associated with a given conformation of a molecule. However, molecular mechanics energies have no meaning as absolute quantities. Only differences in energy between two or more conformations have meaning. A simple molecular mechanics energy equation is given by:
Energy =
Stretching Energy +
Bending Energy +
Torsion Energy +
Non-Bonded Interaction Energy
These equations together with the data (parameters) required to describe the behavior of different kinds of atoms and bonds, is called a force-field. Many different kinds of force-fields have been developed over the years. Some include additional energy terms that describe other kinds of deformations. Some force-fields account for coupling between bending and stretching in adjacent bonds in order to improve the accuracy of the mechanical model.
The mathematical form of the energy terms varies from force-field to force-field. The more common forms will be described.
Stretching Energy
The stretching energy equation is based on Hooke's law. The "kb" parameter controls the stiffness of the bond spring, while "ro" defines its equilibrium length. Unique "kb" and "ro" parameters are assigned to each pair of bonded atoms based on their types (e.g. C-C, C-H, O-C, etc.). This equation estimates the energy associated with vibration about the equilibrium bond length. This is the equation of a parabola, as can be seen in the following plot:
Notice that the model tends to break down as a bond is stretched toward the point of dissociation.
Bending Energy
The bending energy equation is also based on Hooke's law. The "ktheta" parameter controls the stiffness of the angle spring, while "thetao" defines its equilibrium angle. This equation estimates the energy associated with vibration about the equilibrium bond angle:
Unique parameters for angle bending are assigned to each bonded triplet of atoms based on their types (e.g. C-C-C, C-O-C, C-C-H, etc.). The effect of the "kb" and "ktheta" parameters is to broaden or steepen the slope of the parabola. The larger the value of "k", the more energy is required to deform an angle (or bond) from its equilibrium value. Shallow potentials are achieved for "k" values between 0.0 and 1.0. The Hookeian potential is shown in the following plot for three values of "k":
Torsion Energy
The torsion energy is modeled by a simple periodic function, as can be seen in the following plot:
The torsion energy in molecular mechanics is primarily used to correct the remaining energy terms rather than to represent a physical process. The torsional energy represents the amount of energy that must be added to or subtracted from the Stretching Energy + Bending Energy + Non-Bonded Interaction Energy terms to make the total energy agree with experiment or rigorous quantum mechanical calculation for a model dihedral angle (ethane, for example might be used a a model for any H-C-C-H bond).
The "A" parameter controls the amplitude of the curve, the n parameter controls its periodicity, and "phi" shifts the entire curve along the rotation angle axis (tau). The parameters are determined from curve fitting. Unique parameters for torsional rotation are assigned to each bonded quartet of atoms based on their types (e.g. C-C-C-C, C-O-C-N, H-C-C-H, etc.). Torsion potentials with three combinations of "A", "n", and "phi" are shown in the following plot:
Notice that "n" reflects the type symmetry in the dihedral angle. A CH3-CH3 bond, for example, ought to repeat its energy every 120 degrees. The cis conformation of a dihedral angle is assumed to be the zero torsional angle by convention. The parameter phi can be used to synchronize the torsional potential to the initial rotameric state of the molecule whose energy is being computed.
