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:

E_total = E_bond+ E_angle+ E_torsion + E_electro +E_vdw

E_bond, 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 S_bond set of pairs of atoms that are connected by chemical bond defined as;

E_bond = Σ _{(i,j) E Sbond} k_ij^b(α_ijk-α_ijk⁰)²

where:

k_ij^b is the force constant

α_ij⁰ is the equilibrium length

α_ij 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.

E_angle, 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 S_angle the triplet of atoms that are connected by two chemical bonds;

E_angle = Σ _{(i,j,k) E Sangle}k_ijk^a(α_ijk-α_ijk⁰)²

where:

k_ijk is the force constant

α_ijk⁰ is the equilibrium angle

α_ijk is the bond angle

E_torsion, 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 S_torsion the quartet of atoms that are connected by three chemical bonds, as follows;

E_torsion = Σ _{(i,j,k,l) E Storsion}k_ijkl^t[1+cos(nα_ijkl- α_ijkl⁰ )]

where:

k_ijkl^t is the force constant

α_ijkl⁰ is the equilibrium angle

α_ijkl is the torsion angle

E_electro, 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 S_electro, the set of pairs of atoms with electrostatic interactions, as follows;

E_electro = Σ _{(i,j) E Selectro} ( q_iq_j)/(e_ijr_ij)

where:

e_ij is a constant

q_iq_j are charges of atoms

r_ij is the distance between atoms

E_vdW, 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 S_vdW, set of atoms with van der Waals interaction, as:

E_vdw = Σ _{(i,j) E Svdw} ε_ij [ (σ_ij/r_ij)¹² - 2(σ_ij/r_ij)⁶]

where:

r_ij distance between atoms

σ_ij distance at Van der Waals energy is minimum

References

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