1. MTEN-690 : Molecular Modeling
Lecture 20
Vikram K. Kuppa
Vikram K. Kuppa
Department of Chemical & Materials Engineering,
Department of Chemical & Materials Engineering,
University of Cincinnati
University of Cincinnati
601J ERC
601J ERC
Ph: 513-556-2059
Ph: 513-556-2059
Vikram.kuppa@uc.edu
Vikram.kuppa@uc.edu
www.uc.edu/~kuppavm
www.uc.edu/~kuppavm
4. Q: How does one deal with a system with a large no of DOFs?
Imagine a case in which a single molecule is of importance, such
as a heavy ion through a sea of water molecules, or a biopolymer
in solution.
Explicit treatment of solvent is wasteful
Timescales of motion are also different
Additionally, implicit models for such motion (not through
vacuum, but through solution) exist.
We can modify the equations of motion, to take into account the
additional contributions to the molecule of interest through the
solvent. Can now explore phase space of molecule better.
Dynamics
5. The motion of the solvent is assumed to be much faster than that
of the large molecule of interest. The solvent relaxes much more
rapidly than the solute.
Hence, the force that the solute feels, in addition to the gradient
in pairwise potential (intermolecular forces, ex. LJ), is a random
fluctuating force, and a drag force that retards its motion.
γ is the friction coefficient, given in terms of the collision
frequency, ζ = γ/m. For spherical particles of radius a in a
medium of viscosity , we have
Modified eqn of motion
( ) ( )
d
m U r sR t
dt
v
v
6 a
m
6. The friction coefficient is also related to the random force R,
which is assumed to be a Gaussian with mean=0, and variance
dependent on particle mass (m), friction (γ) and temperature.
So, Langevin Equation :
We have
This is a stochastic differential equation. The motion of the
particles is not necessarily physical at short times, but gives an
average diffusion coefficient at long times. D is estimated by
Modified eqn of motion
( ). ( `) 6 ( `)
B
R t R t k Tm t t
( ) ( )
d
m U r sR t
dt
v
v
6 B
s k Tm
/
B
D k T m
7. The random force is uncorrelated at long times.
The relationship b/w γ and R is used to maintain the temperature.
This relationship is dictated by the “Fluctuation-Dissipation”
theorem, which essentially states that the random forces acting
on the particle counteract the effect of the dissipative force due
to friction.
Remember that the friction force acts opposite to the direction of
motion, tending to slow down the particles, while the random
force gives an arbitrary impetus to the particle, speeding it up.
Thus, the friction and the noise are related.
These two effects cancel each other out.
Langevin Equation
8. Neither total energy nor momentum is conserved (think back to
stochastic temperature control, Anderson thermostat)
The random force R thus introduces a stochastic component to
the trajectories: the simulation is not completely deterministic (as
it would be for a purely Newtonian system). Traditional MD is
recovered, as γ goes to 0.
The above equations of motion can be integrated with any of the
standard algorithms discussed earlier.
The Langevin eqn can be extended to include memory effects, i.e.
there is a timescale for the decay of the friction term; and
coupling effects, i.e. the friction depends on the position of the
particle, and its neighbors. Thus, hydrodynamic coupling can be
included (DPD).
Implications
9. If the change in the molecular potential become negligible at long
times, then there is no acceleration of the particle. So, we get
non-inertial dynamics.
So we can get velocities
Of course, we know that the position is related to the time-step
and velocities by
And so
Where the relation b/w D and ζ holds.
Brownian Dynamics
0 ( ) ( )
U r sR t
v
( ) ( )
F r sR t
v
( ) ( )
r t t r t t
v
2
( ) ( )
0; 2
F
r t t r t t
m
D t
r
r r
10. The positions of the particles are determined from the
intermolecular potential, and a random number r
The system is now highly stochastic, and is in fact much closer to
MC than MD. The dynamics correspond to the overdamped limit,
i.e. for strong friction. Recall that this happens at long(er) times,
and for highly viscous media, where the friction is large. This is
especially useful for describing long molecules (polymers, DNA,
proteins)
Neither BD, nor LD as written in their simplest forms are useful
for studying hydrodynamic interactions, since momentum is not
conserved, and the spatial correlations are not considered. For
hydrodynamics, additional method of dissipative particle
dynamics (DPD) is used, which reproduces the Navier-Stokes
equation at large times.
Brownian Dynamics
11. Methods are useful for studying systems with energy barriers, ex.
nucleation, crystallization, etc. The random “kick” due to the
solvent is capable of moving the system over the barriers.
Faster than regular MD, due to coarse-graining of solute degrees
of freedom. So, the phase space of the molecule of interest is
explored in greater detail.
Assigning realistic times is a problem, due to the random nature
of the dynamics. For ex. One cannot directly assign a nucleation
rate for the problem discussed above.
Neither momentum nor energy conserved.
Implications
