# Potential energy of protein

## Intramolecular potential energy

Internal potential energy of a molecule in molecular mechanics is described by a system of functions known as forced field. The potential energy usually includes several terms:

Etotal = Ebond+ Eangle+ Etorsion + Eelectro +Evdw

## Ebond, bond length potential energy contribution

When two atoms are connected by a chemical bond, they tend to maintain a fixed distance. The fixed distance depends on the atoms that forms the bond. Any variation from this fixed distance which is the equilibrium point adds additional potential energy to the protein. This is similar to the concept of Hooke's Law, so the bond can be envisaged as a spring connecting the atoms together. Total potential energy based on bond length is over set Sbond set of pairs of atoms that are connected by chemical bond defined as;

Ebond = Σ (i,j) E Sbond kijbijij0)2

where:

kijb is the force constant

αij0 is the equilibrium length

αij is the current length

for the bond between ith and jth atoms.

Just as the potential energy can be written as a quadratic form in the internal coordinates, so it can also be written in terms of generalized forces. The resulting coefficients are termed compliance constants.

## Eangle, bond angle potential energy contribution

Like bond length, when three atoms are connected with two chemical bonds, the two bonds tend to form a fixed angle. Any variation from this fixed equilibrium angle contributes to protein potential energy. Total potential energy based on angle is defined as over set Sangle the triplet of atoms that are connected by two chemical bonds;

Eangle = Σ (i,j,k) E Sanglekijkaijkijk0)2

where:

kijk is the force constant

αijk0 is the equilibrium angle

αijk is the bond angle

## Etorsion, torsion angle potential energy contribution

The middle bond of three bonds formed by four atoms maintains a certain angle which is also known as torsion is defined over set Storsion the quartet of atoms that are connected by three chemical bonds, as follows;

Etorsion = Σ (i,j,k,l) E Storsionkijklt[1+cos(nαijkl- αijkl0 )]

where:

kijklt is the force constant

αijkl0 is the equilibrium angle

αijkl is the torsion angle

## Eelectro, electrostatic potential energy contribution

Interaction between charged atoms adds potential energy to the protein based on distance of the distance between pairs of atoms defined over Selectro, the set of pairs of atoms with electrostatic interactions, as follows;

Eelectro = Σ (i,j) E Selectro ( qiqj)/(eijrij)

where:

eij is a constant

qiqj are charges of atoms

rij is the distance between atoms

## EvdW, Van der Waals potential energy contribution

Depending on van der Waals radii of atoms every atom of protein interacts with each other that are not far apart. The potential energy contribution of this interaction is defined over SvdW, set of atoms with van der Waals interaction, as:

Evdw = Σ (i,j) E Svdw εij [ (σij/rij)12 - 2(σij/rij)6]

where:

rij distance between atoms

σij distance at Van der Waals energy is minimum

## References

• Wu, Zhijun. Lecture notes on computational structural biology, World scientific publishing, 2008.