Download Incorporating Solvent Effects Into Molecular Dynamics:

Incorporating Solvent
Effects Into Molecular
Dynamics:
Potentials of Mean Force
(PMF) and Stochastic
Dynamics
Eva Zurek
Section 6.8 of M.M.
Introduction:
• Multi-Molecule MD Calculation: simulation of liquids via
placing 10-1000 molecules in a cubic box with periodic boundary
conditions.
• Single-Molecule MD Calculation: consideration of one molecule
in a box which is surrounded by identical copies of itself (usually
done in vacuo).
• Problem: how to incorporate solvation effects into singlemolecule MD calculations without significantly increasing
computational time?
Possible Solutions:
• Langevin Approach: a set of 3N coupled non-stochastic
equations is made stochastic via the addition of a randomly
fluctuating force.
• Fokker-Planck Approach: start from the N-particle phase space
probablility distribution. Solved equations give the probability
distribution directly. Equivalent to Langevin approach.
• Master Equation Approach: start from the N-particle phase
space probablility distribution, which obeys a master equation.
Approach is more general.
The Langevin Equation:
• The most simple form of SD (Stochastic Dynamics) is represented
by the ordinary or strict Langevin Equation (also known as
Brownian Dynamics, BD).
mivÝ
i (t)  mi i vi (t)  Fi {x i (t)} Ri (t)
Collision frequency.
Related to the friction
Systematic Force.
coefficient.
Simulates drag of the To be derived from PMF
motion of the solute (Potential of Mean Force).
through the solvent.
(1)
Random Force.
(random collisions
between solute &
solvent).
The Collision Frequency ():
Ffri ctional  xv
(2)
  x /m
(3)
x  k BT / D
(4)
Ffri ctional  6 av
(5)
• Ffrictional arises from the motion of the particle through the solvent and is
equivalent to the frictional drag.
• It is proportional to the speed of the particle.
• Here x is the friction coefficient, m is mass, v is velocity, D is the diffusion
constant, a is the radius, n is the viscosity of the fluid. Eqs. 4 and 5 apply
only to spherical particles.
• One can consider -1 as being the velocity relaxation time (time taken for
particle to lose memory of its initial velocity)
Potential of Mean Force (PMF):
• A potential that includes the average interaction with the solvent.
(The averaged effects of the solvent on the solute).
• Can be derived from a multi-molecule MD or Monte-Carlo
simulation or from analytical theories of the liquid state.
• Example: 1,2-dichloroethane in the gas phase contains 77% trans
and 23% gauche confomers. In the liquid phase these numbers
become 44% and 56%, respectively. The PMF would be designed
to reproduce this new population and to enable a single 1,2dichloroethane molecule to be simulated as if it were present in the
liquid.
Assumptions about the Random Force:
Ri (0)R j (t)  2mikT i ij (t)
(6)
stationary, markovian

w Ri  2 Ri2


1
2

exp  Ri2 / 2 Ri2

Ri = 0
(7)
(8)
gaussian, with zero mean
vi (0)R j (t) = 0,
t 0
(9)
Fi (0)Rj (t) = 0,
t0
(10)
have no correlation with prior velocities
nor with systematic forces
Algorithms: Dt<<1
• The systematic force is approximated by a constant force, F
during Dt.
• In this case the Langevin equation can be integrated directly
giving:
1
xn 1  x n  vn Dt  an (Dt)2
2
vn1  vn  an Dt
where
an   vn  m 1 (Fn  Rn )
• Here Rn is the average random force over Dt. It is taken from a
gaussian random number generator.
Algorithms: no restrictions on Dt
• The systematic force is approximated by a constant force, F during Dt.
• Integration of the Langevin equation yields very long and ugly
expressions for xn+1 and vn+1.
• Both contain an integral over R, which is a random function obeying
the aforementioned properties. Thus it’s integral obeys a bivariate
gaussian probability distribution, w(xn, vn, Fn| xn+1, vn+1, Dt). The
integral can be obtained by sampling from this distribution.
• It is difficult to properly sample this distribution, so sample from two
univariate distributions instead. First sample xn+1 from a position
probability distribution and then vn+1 from a conditional velocity
probability distribution.
Algorithms: Dt>>1
• The systematic force is approximated by a constant force, F
during Dt.
• The time scale is much larger than the velocity relaxation time.
• This algorithm applies only to systems in the diffusive regime for
which the velocity distribution relaxes much more rapidly than the
position distribution.
• D is the diffusion constant and Xn is a gaussian distribution.
xn 1  x n  Fn (m ) 1 Dt  Xn (Dt)
where
Xn  0
Xn2  2kT(m )1 Dt  2DDt
Other Stochastic Dynamics:
• Time Correlations: are taken into account in Generalized
Langevin Dynamics (GLD). Here the collision frequency, i, (and
thus frictional coefficient) has become time dependent. Computer
simulations of GLD have been confined to gas-solid collisions and
gas phase molecule collisions.
t
mivÝi (t)  mi   i (t' )vi (t  t' )dt' Fi{xi (t)}  Ri (t)
o
• Spatial Correlations: Here the collision frequency, i, (and thus
frictional coefficient) has become space dependent. Has been used
to simulate dimers, trimers and binaphthyl.
3N
3N
j 1
j 1
mivÝi (t)   mi   ij v j (t)  Fi{xi (t)}    ij R j (t)
Practical Aspects I:
• Assignment of :
 For simple spherical particles this may be related to the diffusion
coefficient of the fluid via Eq. 4.
 For a rigid molecule it may be possible to derive  from a standard
MD simulation.
 In general,  is different for all atoms. For simple molecules (ie.
butane)  may be considered the same for all atoms.
 May be obtained from trial and error (performing simulations for
different  and comparing with experiment or MD simulations.)
Practical Aspects II:
• Dramatic time savings can be obtained as opposed to a full MD
calculation. This is due to:
– Much fewer molecules present
– Longer time steps may be used
• MD and SD simulations were performed on Cylosporin in CCl4
and water. The results showed that in water the SD simulation
resulted in an excessive amount
of internal hydrogen bonding
and little hydrogen bonding
with the solvent.
Me
C
N
C
H
O
Me
C
N
C
H
O
Me
CHOH
O
C
N
C
H
C
CH
CH
Me
O
N
C
O
C
N
Me
CH2
CH2
C
HN
C
Cylosporin
NH
O
CH
N
H
C
O
CH
N
C
Me
O
CH
N
H
C
O
C
H
O
References:
• van Gunsteren, W.F.; Berendsen, J.J.C.; Rullmann, J.A. Molecular
Physics, 1981, 44, 69-95.
• van Gunsteren, W.F.; Berendsen, J.J.C. Molecular Physics, 1982,
45, 637-647.
• M.M. Chapter 6.8