Force Fields

The potential function in equation 2.5 and associated parameters are known as a force field. Force fields can be created to describe a system at various levels of detail. Atomistic force fields model each atom in the system as an individual entity, whereas coarse grained force fields model small groups of atoms as individual ”pseudo-atoms” or ”beads”. Atomistic force fields comprise a complete set of parameters and potentials to describe all of the interactions between all of the atoms in the system. Many different force fields have been designed to describe all manner of systems. Key differences in these force fields arise from the method by which the parameters are derived and variations in the potentials used. It is therefore vital to choose an appropriate force field that correctly reproduces known properties of the system of interest [34].

The overall potential in any force field is given by the sum of the bonded and non- bonded interaction potentials (equation 2.13).

Etotal= X

Ebonded+ X

Enon-bonded (2.13) The bonded interactions are given by equation 2.14 and the non bonded interactions are given by equation 2.15.

Ebonded= X Ebonds+ X Eangles+ X Edihedrals (2.14) Enon-bonded= X Eelectrostatic+ X

Evan der waals (2.15)

Bonded Interactions

The bonded interactions in equation 2.14 are the method by which the force field imposes the correct molecular geometry on the system. These are important to implement correctly as the structural characteristics of individual molecules can be influential in the overall behaviour of the system. Ebonds is the bond stretching potential, which acts between two atoms directly connected by a bond and is given by a harmonic potential (equation 2.16) where rij is the distance between two atoms i and j (the bond length) and r0 is the equilibrium bond length. The minimum of this potential corresponds to the equilibrium bond length.

Ebonds(rij) = 1

2K(rij − r0)

2 _(2.16)

The second bonded interaction, Eangles is the angle bending potential, which controls 29

2. Methodology and Technical Details

the behaviour of the angles formed between three adjacent bonded atoms (Θijk). These are represented by another harmonic potential (equation 2.17) where KΘ is a force constant and Θ0 is the equilibrium bond angle.

Eangles(Θijk) = 1

2KΘ(Θijk− Θ0)

2 _(2.17)

For every 4 adjacently bonded atoms in a molecule there is a pair of intersecting planes, each defined by 3 atoms, with 2 atoms in common. The angle between these 2 planes is referred to as the dihedral or torsion angle. In a force field this is known as a proper dihedral and in GROMACS these are commonly defined by the Ryckaert-Bellemans function (equation 2.18) where φijkl is the dihedral angle between the planes defined by atoms ijk and jkl.

Edihedrals(φijkl) = 5 X

n=0

Cn(cos(φijkl))n (2.18) This is known as a proper dihedral and defines the potential energy for rotation around a chemical bond (the bond between atoms j and k) which essentially controls how flex- ible the molecule is. An additional contribution to the overall dihedral term is given by equation 2.19 which is known as the improper dihedral potential. Improper dihedrals are dihedrals where the 4 atoms defining the planes are not all explicitly linked by bonds and are often used to enforce properties such as the planarity of aromatic rings. In the potential form given in equation 2.19, ϕijkl is the angle between the ijk and jkl planes, n is the periodicity and ϕ0 is the angle corresponding to the potential maximum. K is once again a force constant.

Eimproper(ϕijkl) = K(1 + cos(nϕijkl− ϕ0)) (2.19) The dihedral term is of particular importance in the context of this thesis as we focus on re-parametrizing this aspect of the force field to better capture peptoid backbone torsions. This is discussed in greater detail in Chapter 4.

Non-Bonded Interactions

In the majority of force fields there are two major components describing the non-bonded interactions in the system. These are the electrostatic and Van der Waals interactions. The electrostatic potential is necessary to define the interactions between the partial charges assigned to each atom in the system. These can be described in a quick and simple manner by the Coulomb interaction (equation 2.20) which is implemented within a specified cut- off distance, beyond which the interaction potential is set to 0. However, this treatment neglects the effect of long range electrostatics, as in reality Eelectrostatic tends to 0 very

2. Methodology and Technical Details

slowly as r increases and therefore reasonable cut-off distances may exclude interactions at larger r that may be important in capturing the correct behaviour of the system.

Eelectrostatic= qiqj

4π0r (2.20)

In equation 2.20 qi and qj are the charges on atoms i and j respectively, r is the distance between the 2 atoms and 0 is the permittivity of free space.

There are several methods designed to improve upon the use of the Coulomb potential and include long-range electrostatics in force fields but perhaps the most popular (and the one implemented in the work in this thesis) is the particle-mesh Ewald (PME) method. In this method, the total electrostatic interactions are split into short-range and long-range components according to a cut-off distance. The short range interactions are described by the Coulomb potential which converges quickly in real space and the long-range interac- tions are calculated by a summation in Fourier space [35].

The Van der Waals interactions are normally described in atomistic force fields as a function of distance, r, by the Lennard-Jones potential (equation 2.21) where ε is the potential well depth and σ is the distance at which the potential is equal to 0.

ELennard-Jones= 4ε "  σ r 12 − σ r 6#