A set of coordinates is required as input into the simulation. The system is then evolved
from the starting coordinates during an equilibration phase where structural and thermo-
dynamic properties are closely monitored until a plateau is reached. After equilibration, a
production run is performed during which simple properties of the system are calculated.
The configuration of the system is saved at regular intervals. Finally, post simulation anal-
ysis is performed and the output configurations studied. Unusual changes in the structure
of the system can highlight abnormalities in the simulation.
2.6.1
The Initial Configuration
Initial coordinates of the system can be taken from experimental data,e.g. NMR or X-Ray
crystallography, generated by theoretical modelling or a combination of both. The choice
of this configuration is critical as this it can determine the success of the simulation. High-
energy interactions, which may cause instabilities in the simulation, can be eradicated by
performing an energy minimisation before the simulation.
2.6.2
Energy Minimisation
Energy minimisation algorithms are used to identify the geometries of a system that corre-
spond to minimum points on the energy surface. Two first-order, first derivative, minimi-
sation algorithms which are commonly used in molecular modelling are Steepest Descent
(SD) and Conjugate Gradient (CG). Both methods gradually change the coordinates of the
system as they move down a gradient bringing the structure closer to the minimum. The
starting set of coordinates for each iteration is the molecular configuration generated by the
previous step.
Steepest Descents moves in the direction parallel to the net force. For3N coordinates the
displacement h0 defines how far to move along the gradient. The forces F and potential energy are calculated and new positions are computed using equation 2.21.
rn+ 1 =rn+
Fn
max (|Fn|)
hn (2.21)
whereFnis the force or negative gradient of the potentialV andhnthe maximum displace-
ment. The largest absolute values of the force components are denoted max (|Fn|). The
forces and potential energy of the new positions are then determined and the new positions
accepted or rejected based on a set of criteria. In GROMACS positions are accepted when
(Vn+1 < Vn) and hn+1 then becomes 1.2 hn. Positions are rejected when (Vn+1 > Vn)
and hnbecomes 0.2 hn. The search ceases when either the number of user specified force
evaluations have been performed or whenmax(|Fn|) is less than a specified value 0.
Steepest Descents is a good method for relieving the highest-energy interactions in an initial
structure, as the direction of the gradient is determined by the largest interatomic forces.
Even when the starting configuration is far from the energy minimum, where the harmonic
approximation of the energy surface is often a poor assumption, this method is very robust.
It is also easy to implement. However, if the minimum is located at the base of a long
narrow valley a vast number of very small steps will be required to obtain convergence. At
each step a right-angles turn is required, generating an oscillating path which continually
overcorrects itself. During the later stages of the minimisation errors corrected by earlier
moves are reintroduced 228.
The conjugate gradient method is slower than the steepest descent in the initial minimiza-
tion stages, but is more efficient the closer the structure is to the energy minimum. The
path generated by the conjugate gradient algorithm in narrow valleys does not exhibit the
oscillatory behaviour of the steepest descents method, as although the gradients of each step
are orthogonol the directions are conjugate ‘M’ steps. This method moves from pointxkin
vk =−gk=γkvk−1 (2.22) whereγkis a scalar constant 231.
A short run of steepest descent minimsation followed by conjugate gradients minimisation
can be used to achieve a relaxed starting structure.
2.6.3
Generating the Initial Velocities
Next, initial atomic velocities, if not available, must be assigned. This is usually done
by randomly selecting from a either a Maxwell-Boltzmann or Gaussian distribution at the
required temperature, corrected so that there is no overall momentum. The Maxwell-
Boltzmann equation, equation 2.23, can be used to obtain the probability density for the
velocity componentvix at given absolute temperatureT for an atomI of mass in the direc-
tionx. p(vi) = s mi 2πkBT exp −miv 2 i 2kBT ! (2.23)
where kB is Boltzmann’s constant. Similar equations apply in they and z directions. A
random seed is used to generate the first set of velocities which for the leap-frog algorithm
are at t = t0 - ∆t/2. In GROMACS normally distributed random numbers are generated by adding twelve random numbersRk in the range 0 =Rk = 1, and subtracting 6.0 from
their sum, this is then multiplied by the standard deviation of the velocity distribution. A
correction is then made to ensure the resulting total energy will correspond to the desired
temperature. The center-of-mass motion is removed and velocities are scaled to ensure the
2.6.4
Equilibration
The starting point for a simulation is often at a different density or temperature to that
required. Therefore, it is necessary to run the simulation for a period to allow the system to
come to equilibrium. At the end of this phase of the simulation all memory of the starting
configuration should have been lost. Various parameters - including the temperature, energy
and pressure of the system in addition to the configuration of system - can be monitored
during this phase. When these parameters cease to exhibit a systematic drift and start to
oscillate about steady mean values the production run can commence.