Incorporating Solvent Incorporating Solvent Effects Into Molecular Effects Into Molecular

Dynamics:Dynamics:

Potentials of Mean Force (PMF) and Stochastic

DynamicsEva Zurek Section 6.8 of M.M.

• 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 single-molecule MD calculations without significantly increasing computational time?

Introduction:Introduction:

• 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.

Possible Solutions:Possible Solutions:

The Langevin Equation: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).

mi ˙ v i (t) =−miγivi(t)+Fi{xi(t)}+Ri(t) (1)

Systematic Force.To be derived from PMF(Potential of Mean Force).

Random Force.(random collisionsbetween solute &solvent).

Collision frequency.Related to the frictioncoefficient.Simulates drag of the motion of the solutethrough the solvent.

The Collision Frequency (The Collision Frequency ():):

• 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 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)

Ffrictional=−ξv (2)

γ =ξ /m (3)

ξ =kBT / D (4)

Ffrictional=6πaηv (5)

Potential of Mean Force (PMF):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 1,2-dichloroethane molecule to be simulated as if it were present in the liquid.

Assumptions about the Random Force:Assumptions about the Random Force:

Ri(0)Rj(t) =2mikTγ iδijδ(t) (6)

stationary, markovian

w Ri = 2π Ri2

( )−1

2 exp−Ri2 /2 Ri

2( ) (7)

R i =0 (8)

gaussian, with zero mean

vi(0)Rj(t) =0, t ≥0 (9)

Fi(0)Rj (t) =0, t≥0 (10)

have no correlation with prior velocities

nor with systematic forces

Algorithms: Algorithms: t<<1t<<1

• The systematic force is approximated by a constant force, F during t.• In this case the Langevin equation can be integrated directly giving:

• Here Rn is the average random force over t. It is taken from a gaussian random number generator.

xn+1 =xn +vnΔt+12

an(Δt)2

vn+1 =vn +anΔt

where

an =−γvn +m−1(Fn +Rn)

Algorithms: no restrictions on Algorithms: no restrictions on tt• The systematic force is approximated by a constant force, F during t.

• 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, t). 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: Algorithms: t>>1t>>1

• The systematic force is approximated by a constant force, F during t.

• 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 =xn +Fn(mγ)−1Δt +Xn(Δt)

where

Xn =0

Xn2 =2kT(mγ)−1Δt =2DΔt

Other Stochastic Dynamics: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.

• Spatial Correlations: Here the collision frequency, i, (and thus frictional coefficient) has become space dependent. Has been used to simulate dimers, trimers and binaphthyl.

mi ˙ v i (t) =−mi γi(t')vi (t−t' )dt'+Fi{xi(t)}+Ri(t)o

t

∫

mi ˙ v i (t) =−mi γijvj(t)+Fi{xi(t)}+ αijRj(t)j=1

3N

∑j=1

3N

∑

Practical Aspects I: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.)

• 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.

Practical Aspects II:Practical Aspects II:

C N

Me

CH

C

O

N CH

Me

C N

O Me

CH

CHOH

C

O

NH

CH

C O

N Me

CH2

C O

CH

C

O

NH

CHC

O

N

Me

CHC

O

NH

CHC

HN

O

CH2

CO

N

CH

Me

Cylosporin

References: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