0.8 Motion planning for proteins: biophysics and applications  (Page 2/10)

Approximate energy functions provide the basis for molecular simulations and some protein-ligand docking procedures, among other applications. In some docking problems, a potential function is used to evaluate how likely a particular pose of a small molecule (ligand) in the binding pocket of a protein is. The internal energy of the receptor and the ligand are considered along with the interaction energy between the two. Interaction energy usually consists of the non-bonded terms found in the internal energy function, summed of all pairs of atoms (r,l), where r is an atom of the receptor and l is an atom of the ligand. If the energy function approximates what is going on well enough then the docked conformation should have minimum energy value. Some docking programs use alternative forms of scoring functions, but in all cases, the object is to find the state of the complex that has the least free energy, and therefore there is a balance between making functions fast to compute and making them reasonably approximate free energy. Potential functions may also be used in simulations to study protein folding mechanisms and kinetics.

Terms of energy functions

Bonds

The bond energy term corresponds to the stretching and compressing of the length of a bond. In most energy functions this term reduces bonds to simple harmonic oscillators, yielding a quadratic equation: $$E_{\text{bonds}}=K_b(b-b_0)^2$$ where $K_b$ is an empirically determined constant that depends on the atom types, $b$ is the current bond's length, and $b_0$ is the bonds length in equilibrium, which again depends on the atom types. In this case you can think of the bond as a spring, it has an equilibrium length that it wants to remain at. If the bond length varies from the equilibrium length, the energy increases.

Bond angles

The bond angle energy corresponds to changes in the angle between bonds. As with bond length, the bond angles have an equilibrium value, and any deviation increases the potential energy. Once again this can be modeled by a simple quadratic term. $$E_{\text{angles}}=K_{\theta }(\theta -{\theta }_0)^2$$ where $K_{\theta }$ is an empirically determined constant, $\theta$ is the current bond angle, and ${\theta }_0$ is the equilibrium angle.

Torsions

Torsions are created by series of three bonds, and consist of rotations of the bonds on either end with respect to the axis of the middle bond. In molecular structure certain torsional angles are preferred over others and the energy function reflects this. Usually it is described by a Fourier series expansion. The simplest being a single term:

A more complicated three term expansion can also be used:

Van der waals interactions and steric clash

Strictly speaking, Van der Waals interactions are weak attractive interactions between atoms at an ideal separation from each other. The atoms transiently induce each other's electron distribution into complementary dipoles, allowing a weak electrostatic attraction between them. In molecular potential fields, Van der Waals attractions are usually combined with steric clash (extremely high energies due to overlapping atoms) in a Lennard-Jones potential, such as this Lennard-Jones 12-6 function: