Computer Simulation Methods 3

Computer simulation
methods
(3)
Dr. Vania Calandrini

in the previous lecture:
setting up a simulation
boundaries
monitoring the equilibration
short-range and long-range interactions
minimum image, cutoff and related problems/solutions
Ewald summation
Particle Mesh Ewald summation

Molecular dynamics
Classical Molecular
Dynamics
technique for computing the equilibrium and transport properties of a
classical many-body system.
the nuclear motion of the constituents particles obeys the laws
of classical mechanics (hυ υ< 6ps -1 at 300 K )
These vibrations are not well treated by
classical physics. They require quantum
mechanics simulations.
These data are used as input in molecular
dynamics simulations to parametrize the
characteristic frequencies of the potential.

In molecular dynamics, successive configurations are generated by integrating Newtonʼs
laws of motion. The result is a trajectory that specifies how the positions and velocities of
the particles in the system vary with time.
1) a body continues to move in a straight line at constant velocity unless a force acts upon it
2) force equals the rate of change of momentum
3) to every action there is an equal and opposite reaction
The trajectory is obtained by solving the differential equations embodied in Newtonʼs second
law:
m i¨r i ≡ ṗ i = − ∂V =Fi = F i
∂r i
ṙ i = p i
m i
conservative system :
each atom is represented
by a mass point
ri
V = V (r 1 , r 2 ,...,r n )

Time and Spatial scales
MD simulations
(atomic details):
Biological
Systems:
~ 10 -12 to 10 -7 s
~ 10 -10 to 10 -8 m
~ 100 000 particles

Molecular dynamics with continuous
Potentials
Under the influence of a continuous potential the motions of all the particles are coupled
together, giving rise to a many body problem that cannot be solved analytically=> then the
equations of motion are integrated numerically
Numerical integration of motion equations : Finite difference methods
the integration is broken down into many small stages, each separated in time by a fixed δt: forces acting on
each particle at time t are computed knowing the positions of all the particles. Then the accelerations together
with the positions and velocities at time t are used to compute the new positions and velocities at time t+δt.
During each time step the force is assumed to be constant.
Taylor series
expansions:
Any finite difference integrator is naturally an approximation for a system developing
continuously in time.

The integrator is responsible for the accuracy of the simulation results.
Main requirements:
• Stable: it has to conserve energy
• Robust, in the sense that it allows for large time steps in order to propagate the system
efficiently through phase space

Verlet algorithm :
adding up (1) and (2):
(1)
(2)
Taylor’s
expansions
Δt→0
or:
next current previous
requires the storage of old and current positions
subtracting (2) from (1) one can derive the velocity from the knowledge of the trajectory:
or:

Verlet algorithm :
requires the storage of old and current positions
the velocities are not available until the positions have been computed at the next step
it is not a self starting algorithm: at t=0 you have only one set of positions. To obtain the
positions et t=-δt, one can use eq. 2 truncated after the first term:
low precision
time reversible = reversing the momenta of all particles at a given instant, the system
trace back its trajectory in phase space

Leap frog algorithm :
positions and velocities not synchronized

velocity Verlet algorithm :
(1)
v(t + ∆t
2 )=v(t)+∆t 2
f(t)
m + O(∆t2 )
Δt/2→0
Taylor’s expansion
(2)
r(t + ∆t) =r(t)+v(t + ∆t
2 )∆t + O(∆t2 )
from the mean
value theorem or
central difference
(3)
v(t + ∆t) =v(t + ∆t
2 )+∆t 2
f(t + ∆t)
m
f(t + ∆t) =−
- Velocity Verlet requires only the storage of current positions.
- Verlet and velocity Verlet schemes produce exactly the same trajectory
approximation of derivatives by finite differences :
forward difference
backward difference
central difference
δV (r(t + ∆t))
δr
+ O(∆t)
from the
backward
difference

High-order schemes: predictor-correctors
The basic idea is to use r(t) and its first n derivatives at time t to make a prediction for
r(t+ Δt) and its first n derivatives at time t+ Δt. We then compute the forces (i.e.
accelerations) at the predicted positions and find that these forces are different from
the predicted ones. So we adjust our predictions for the accelerations to match the
facts. Moreover, on the basis of the observed discrepancy between the predicted and
observed accelerations, we also try to improve our estimate of the positions and the
remaining n−1 derivatives. the “recipe” used in applying this correction is a
compromise between accuracy and stability.
see Understanding Molecular Simulation by Daan Frenkel
and Berend Smit for some specific examples of a predictor
corrector algorithm.

Comments:
• Although, computational speed seems important, it is usually not very relevant, because
the fraction of time spent on integrating is usually small (as opposed to computing
interactions) at least for simple molecular system.
• Accuracy: we can consider the integration error as the source of the divergence between
the “true” trajectory and the computed trajectory compatible with the same initial conditions
(note that this problem need not to be serious in molecular dynamics (MD) simulations
since the aim of an MD simulation is to predict the average behavior and not what will
precisely happen to a system in a known initial condition. It has been proven that MD
simulations provide good statistical predictions for many systems.)
• Algorithms that allow the use of large time steps require the storage of increasingly higher
order derivatives.
• Energy conservation: higher order algorithms tend to have very good energy
conservation for short times, while the overall energy drifts for long times. In contrast Verlet
algorithms tend to have only moderate short term energy conservation but little long-term
drift.
•Time reversibility (= reversing the momenta of all particles at a given instant, the system
trace back its trajectory in phase space): many algorithms such as predictor-correctors are
not time reversible.Verlet scheme is time reversible.

Tutorials

Required programs:
NAMD
VMD
a parallel molecular dynamics code designed for high-performance simulation of
large biomolecular systems.
VMD is a molecular visualization program for displaying, animating, and analyzing
large biomolecular systems using 3-D graphics and built-in scripting. It can be used
to setup simulations and analyze trajectories .
by the Theoretical and Computational Biophysics Group at the Beckman Institute for
Advanced Science and Technology of the University of Illinois at Urbana-Champaign
you need also a text editor and a plotting program

Tutorial
http://www.ks.uiuc.edu/Training/Tutorials/namd-index.html
NAMD Tutorial
the examples in the tutorial will focus on the study of a small protein, ubiquitin
purpose:
- basic steps of a molecular dynamics simulation, i.e., preparation, minimization, and
equilibration of your system.
- typical simulation techniques and analysis of equilibrium properties
- descriptions of all files needed for the simulations

To do:
do this analysis for
water box

careful
reading

in this lecture:
Molecular dynamics with continuous Potentials:
time and spatial scales
finite difference methods:
Verlet algorithm
velocity Verlet algorithm
Leap frog algorithm
Tutorial (using NAMD and VMD):
basic steps of a molecular dynamics simulation, i.e., preparation, minimization, and
equilibration of the system and analysis of equilibrium properties

for any question:
v.calandrini@grs-sim.de