The bond-stretching and angle-bending terms, discussed above, are often referred to as “hard” degrees of freedom since large energies are required to cause significant deviations from the equilibrium geometries. Most of the complex variations in structure and relative energies observed in biological systems are due to the “softer” torsional and non-bonded contributions.

The barriers to rotation about a bond can be modeled in one of two ways. In very early force fields it was believed that rotational barriers could be omitted. The gauche-trans energy differences would be reproduced by the non-bonded interactions. However, it was quickly realized that for organic molecules neglecting dihedrals made successful parameterization of force fields, to reproduce experimental observables, an almost impossible task and so dihedral terms were explicitly included. The AMBER force field, in common with a number of other biologically oriented force fields, uses a Fourier series expansion for the torsional potential

N у

У(Ф) = £ у [1 + cos(^ — 7)] (8.4)

n=0 2

where Vn is the relative barrier height to rotation, n is the multiplicity (number of minima in a 360° rotation), ф is the dihedral angle, and у is the phase factor which determines the location of the minima. Vn is often termed the relative barrier height since other terms in the force field equation contribute to the barrier height as the bond is rotated, especially the 1-4 non-bonded interactions discussed below. The advantage of using a Fourier series expansion for the dihedral terms centers on the fact that terms of differing multiplicity can be combined to describe complex torsional profiles (Figure 8.2).

Improper torsion angles, also known as out-of-plane bending, are defined for four atoms that are not bonded in a serial manner. They are used to maintain planarity where necessary. The AMBER force field accounts for improper torsions in the same way as regular torsion angles but using a twofold multiplicity.