Force fields and topologies

In addition to the coordinate file, which describes the positions of the atoms, we also need information on how atoms interact with each other and how they act as a function of time as a result. The interactions are calculated through forces that atoms have on each other. The forces in turn are calculated from a potential energy function according to basic Newtonian mechanics. This potential energy function V and the derived mathematical equations describing how the atoms of a system interact with each other is known as the force field. The force field also includes other relevant parameters such as cut-off distances of interactions.

Equations of motion

The potential functionV( r₁,r₂, ...,r_N)represents the potential energy of a system ofN interacting atoms as a function of their position r_i = (x_i,y_i,z_i). The forceF_i applied on a given particleican be calculated from the potential function as

F_i=−dV

dr_i. (5.1)

Once we have calculated the forces applied on alliparticles of the system, we can use them to solve the Newton’s equations of motion

m_id²r_i

d t² =F_i, (5.2)

from which we can integrate the motions of the atoms with an algorithm of our choice[139]. By repeating the previous steps, we can see how the system evolves with time.

The most common algorithm for calculating the equations of motion is the leap- frog algorithm, which is a third-order numerical integration method[140]. It has

several major advantages in mechanics problems including time reversibility and subsequently the conservation of the equation of the system. Its equations can be written as

v_i

t+1 2Δt

=v_i

t−1 2Δt

+F_i(t)

m_i Δt, (5.3)

r_i(t+Δt) =r_i(t) +v_i

t+1 2Δt

Δt, (5.4)

wherev_i is the velocity of the particle, r_i is its position, andΔt is the length of the time step.

The coordinates and velocities at each time step can be saved to a trajectory file, which can then be analyzed to give a detailed look on the behavior of the system over time. The time reversibility is a product of the fact that the positions and the velocities are calculated in a symmetrical way – one can always return to a previous time step by reversing the algorithm.

Potential function

The potential function is typically divided into two categories – bonded and non- bonded interactions – depending on whether the two atoms are connected by a chem- ical bond or if they only interact through long-range interactions[141].

The bonded interactions include the stretching of covalent bonds, the vibration of bond angles, and proper and improper dihedral angles, which describe the angles between adjacent planes and whether the molecules around the bond stay in their cis- ortrans-configurations.

The potential function of a covalent bond is typically described by a harmonic potential

V_bond r_{i j}

=1 2k_{i j}^r

r_{i j}−r_{i j}⁰₂

, (5.5)

wherei and j are the two bonded atoms,k_{i j}^r the force constant of the bond,r_{i j} the distance between the two atoms, andr_{i j}⁰ the reference distance, where the energy of the bond is the lowest. Small deviations from the reference distance increase the potential energy and cause harmonic stretching.

The angles between two covalent bonds are also susceptible to change and this is typically represented by another harmonic potential function. The harmonic po- tential for bond angle vibration on a triplet of connected atomsi–j –kis described by

V_angle θ_{i j k}

= 1 2k_{i j k}^θ

θ_{i j k}−θ⁰_{i j k2}

, (5.6)

whereθ_{i j k}is the angle between the bondsi–jand j–k,k_{i j k}^θ is the force constant of the angle, andθ⁰_{i j k}is the reference angle, where the potential energy is the lowest.

In order to define internal rotations, a function for torsional dihedral angle poten- tial is also required. These are referred to as potential energy functions for dihedral angles. A typical way to define such a function for four connected atomsi– j –k – l is by a periodical function

V_dihedral φ_{i j k l}

=k_φ

1+cos

nφ_{i j k l}−φ_s

, (5.7)

where nis an integer defining the periodicity, k_φ the force constant,φ_{i j k l} the angle between the two planesi j kand j k l, andφ_sthe reference angle. If the atoms i and l are on the same side of the bond j – k, the dihedral angle is zero and it corresponds to acisconfiguration. Likewise, if they are on the opposite sides of the bond, the dihedral angle is 180 degrees, which corresponds to atransconfiguration.

In order to force planar atom groups to stay planar, or to prevent molecules from flipping to their mirror images, a group of improper dihedral functions can be de- fined. They can be periodic, in which case their form is identical to the proper peri- odic dihedral, or more usually of a harmonic type, where their potential function is defined by

V_id ξ_{i j k l}

=1 2k_ξ

ξ_{i j k l}−ξ_{i j k l}⁰ ₂

, (5.8)

whereξ_{i j k l} corresponds to the angle between the planesi j kand j k l,ξ_{i j k l}⁰ is the reference angle, andk_ξ is the force constant.

The non-bonded interactions of a force field are calculated as the sum of the elec- trostatic interactions and the van der Waals forces between two interacting atoms.

They are long-range interactions and thus, unlike in bonded interactions, these atoms can interact with thousands of other atoms in the system.

The electrostatic term is depicted with the Coulombic potential

V_C r_{i j}

= q_iq_j

4πε₀ε_rr_{i j}, (5.9)

whereq_iandq_jare the charges of the two interacting atoms,ε₀is the permittivity of the vacuum,ε_r is the relative permittivity of the medium, andr_{i j} is the distance between the two atoms.

The van der Waals forces are a sum of an attractive and repulsive term, usually depicted with the Lennard-Jones potential

V_LJ r_{i j}

=4ε_{i j} σ_{i j}

r_{i j 12}

− σ_{i j}

r_{i j 6}

, (5.10)

where ε_{i j} is the minimum energy of the potential well and σ_{i j} is the van der Waals distance, defining the point where the potential energy turns from repulsive to attractive.

Calculating all of the non-bonded interactions between every atom pair of the sys- tem at each time step would be computationally extremely heavy. However, as the strength of long-range interactions weakens as the distance r_{i j} between two atoms increases, it may be possible to use a certain cut-off radius to speed calculations up.

The Lennard-Jones potential has an extremely fast decay rate, so short cut-off radii (of around 2 nm) can be used without creating unnecessary artifacts.

On the other hand, the Coulombic potential does not decay as rapidly, so a sin- gular cut-off radius would produce inaccurate results. One typically used method for approximating long-range Coulombic interactions is the Particle Mesh Ewald method (PME), which greatly improves the calculation rates of these interactions [142]. There we employ a cut-off range after which the electrostatic energies are cal- culated in an efficient way with the Ewald method by utilizing Fourier transforms into reciprocal space and back with the help of periodic boundary conditions. PME is an improvement of the basic Ewald method, using a charge grid in the reciprocal space, improving both accuracy and computational speed.

Temperature and pressure coupling

After creating a system, it can be studied in the microcanonical ensemble, where the number of particles, the volume and the total energy are constant. While this ensem- ble is appealing, it can, however, be problematic since quite often the total energy of the system studied in experiments is not conserved. If one would like to compare the results of simulations to the results of experiments as directly as possible, then one obvious way is to carry out the simulations at constant temperature. In our simulations we used this approach, that is, the NpT ensemble with a set number of particles(N), pressure(p)and temperature(T). In order to keep these conditions close to the desired value, external barostats and thermostats are included. These are algorithms which control the pressure and temperature of the system preventing them from drifting due to external forces or computational errors.

The temperature coupling in this work was performed with the velocity rescale thermostat, which is a modified version of the Berendsen thermostat [143]. The Berendsen algorithm uses an external heat bath set at a reference temperature T₀, and corrects the deviation of the system temperature as

d T

d t =T₀−T

τ_T , (5.11)

where T is the temperature of the system and τ_T is the time constant for the exponential decay of the temperature deviation. The strength of the coupling can be adjusted by changing the time constant, which can be useful, for example, for a quick equilibration of the system.

The velocity rescale thermostat differs from the Berendsen thermostat only by an additional stochastic term in its algorithm for the kinetic energy of the system.

The function of this additional term is to ensure correct kinetic energy distribution in the system in order to produce a correct canonical ensemble.

Several different methods also exist for controlling the pressure of the system.

We used the popular Parrinello-Rahman barostat[144], which forces the box vectors (vectors determining the size of the simulation box) denoted by a matrixbto follow an equation of motion

db²

d t² =VW⁻¹b⁻¹(P−P_ref), (5.12)

thus controlling the pressure matrix P by changing the volume V of the box.

The pressure is directed towards the reference pressureP_refby the matrix parameter W, which defines the strength of the coupling. The coupling parameter is defined as

W⁻¹

i j= 4π²β_{i j}

3τ²_pL , (5.13)

where β_{i j} are the isothermal compressibilities, τ_p the pressure time constant, and Lthe largest box matrix element. The pressure is presented as a 3×3 tensor, which allows non-isotropic handling of the system pressure, which is desirable in the case of lipid bilayers. The Parrinello-Rahman barostat also produces a true NpT ensemble.

Periodic boundary conditions

To minimize the computational costs of a large system, molecular dynamic simula- tions are typically conducted in a finite box. In order to avoid artifacts caused by the sides of the box, periodic boundary conditions (PBCs) are often introduced in the simulations. Periodic boundary conditions place an infinite number of identical copies of the simulated system around itself. For bilayer systems, this basically ex- tends a bilayer placed on thexy-plane to infinity. As a result, every particle in the simulated box can interact not only with the other particles placed in the same box, but also their images projected to adjacent boxes. In a similar fashion, when a parti- cle crosses the box boundary, a replica of the same particle appears on the opposite side of the original box. As a result, the amount of particlesNin the system remains constant.

Due to the nature of periodic boundary conditions, the simulation box has to be large enough to avoid artifacts that could arise from interactions with the images of particles. The size requirements are thus dependent on the employed cut-off val- ues for non-bonded interactions. Even though the original box is copied numerous times, only the properties of the original simulation box need to be calculated. This makes PBCs a fast and reliable method for handling the boundaries of a simulation box.