Bonded interactions

Bond stretching

A harmonic potential is the simplest model for bond stretching. It is commonly used in sim- ulations of biomolecular and polymeric systems, although the so-called FENE potential [144] (discussed below) has become increasingly common. The harmonic bond-stretching potential (see Fig. 4.1) between two atomsiand jcan be written as

U(ri j)= 1

2ki j(ri j−ri j,0)

2_{, (4.16)}

whereri j is the bond length with its minimum atri j,0. ki j is the force constant. This approx- imation is simply a Taylor expansion about the equilibrium ri j,0 distance truncated after the

The justification for the use of harmonic term has to be considered carefully: if deviations fromri j,0 become large, the harmonic approximation breaks down. An obvious improvement

is to include higher order terms.

To get a more accurate model, cubic or quartic terms may be included in the Taylor expan- sion (see Fig. 4.1),

U(ri j)= 1 2 h ki j+k_{i j}(3)(ri j−ri j,0)+k(4)i j (ri j−ri j,0) 2i (ri j−ri j,0)2. (4.17)

The force constant for the cubic term is negative to correct for the harmonic potential which is too strong for long bonds – the forces resulting from stretching the harmonic potential are very large. Inclusion of cubic terms allows for dissociation but may easily lead to unwanted dissociation. The MM2 [145] force field includes a cubic terms. Quartic term makes the potential anharmonic and asymmetric, thus reflecting better the the behaviour of real bonds. The MM3 [146, 147, 148, 149] force fields includes quartic terms.

The Morse potential [143] (see Fig. 4.1) is often used to model diatomic molecules and occasionally atom-surface interactions. The differences to the harmonic potential are the fol- lowing: 1) the Morse potential is asymmetric, it is harder to push the two atoms towards each other than to pull them apart, 2) the Morse potential is anharmonic like real bonds are, 3) it includes dissociation, i.e., bond breaking, since it levels instead of diverging as the distance between the two atoms increases. The functional form is given as

UMorse(ri j)= Di j

h

1−exp(−αi j(ri j−ri j,0))

i2

(4.18) ri j is the distance between atomsiand j,ri j,0 is the corresponding equilibrium distance,Di j is the dissociation energy of the bond (i.e., it gives the depth of the potential well), andαi j is an experimental parameter controlling the width of the potential well (smaller value corresponds to a wider potential well). By Taylor expandingUMorseand comparing the terms with Eq. 4.17

it can be seen thatαi jcan be related to the force and dissociation constants asαi j =

√

ki j/2Di j. The FENE potential [144] (see Fig. 4.1) is defined as

U(ri j)=− 1 2kR 2 ln       1 − r 2 i j R2.        (4.19)

At short distances it behaves like a harmonic potential (Eq. 4.16) with force constant k and diverges to positive infinity at the distanceR, preventing bonds from being stretched beyond a maximum lengthR. It is anharmonic and asymmetric and it is used to describe elastic bonds in the bead-spring model for polymers.

Angle bending

Like bond stretching, angle bending can be modelled by polynomial expansions. The most commonly used is the harmonic potential

U(θi jk)= 1

2ki jk(θi jk−θi jk,0)

2_.

Figure 4.1: Plot of the commonly used potentials to model bond stretching.

θi jkis the angle between the three atomsi, jandk,θi jk,0is the value of the angle at equilibrium

andki jk is the force constant.

Similar to bond stretching higher order terms can be included if the angles might deviate much from their equilibrium values. There are certain limitations to the harmonic potential, for example the description of a linear bond angle or systems with more than one equilibrium value for the bond angle; in the latter case a periodical potential function similar to the potentials for bond rotation described below can be used.

Bond rotation (torsion angle)

Torsion angle rotations are very important in biomolecular modeling. In the two above interac- tions, we used the harmonic approximations. That was justified by assuming small deviations

from the equilibrium positions. Here, however, that is not the case since the energy barriers for rotation around a single bond are low and thus allow for large changes. We must also account for multiplicity, i.e., the potential has 2πperiodicity. Instead of Taylor series, cosine expansion is the most common approach. The reason for this choice of an expansion of periodic functions such as cosine functions is the periodicity of the torsion potential itself. For four atoms linked in the orderi− j−k−l, the torsion angleφi jkl is the angle betweenk-land the plane spanned byi-j-k(see Fig. 4.2).

Figure 4.2: Definition of the torsion angleφ: the angle betweenk-land the plane spanned by i-j-k.

Using a cosine expansion, the functional form is

U(φi jkl)= 1 2 X {n}i jkl Vn,i jkl h 1+(−1)n+1cos(nφi jkl+ Ψn,i jkl) i (4.21)

for−π < φi jkl ≤ π. Vn,i jkl is the amplitude, {n} is the set of periodicities andΨn,i jkl the phase angle and is useful in systems with large stereoelectronic effects, i.e. the structure of a molecule being influenced due to an electronic effect. Most of the torsion potentials are at their minimum in thetransconformation (i. e. torsion angles from±5/6πtoπ), thus the factor (−1)n+1ensures that at φ = ±π andφ = 0 the sum is zero. The definition oftransandgauchefor the torsion angle is shown in Fig. 4.3.

Figure 4.3: The definition oftransandgauchefor the torsion angle.