332 8.2  Molecular Simulation Methods

then this can be a physically accurate model for several important properties. However, even

so, no implicit solvent model can account for the effects of hydrophobicity (see Chapter 2),

water viscous drag effects on the biomolecule, or hydrogen bonding within the water itself.

8.2.7  LANGEVIN AND BROWNIAN DYNAMICS

The viscous drag forces not accounted for in implicit solvent models can be included by using

the methods of Langevin dynamics (LD). The equation of motion for an atom in an energy

potential U exhibiting Brownian motion, of mass m in a solvent of viscous drag coefficient

γ, involves inertial and viscous forces, but also a random stochastic element, known as the

Langevin force, R(t), embodied in the Langevin equation:

(8.26)

mr

r

U r

k TmR t

B





+

= −∇( )+

( )

γ

π

2

Over a long period of time, the mean of the Langevin force is zero but with a finite variance of

2πk Tm

B

. Similarly, the effect on the velocity of an atom is to add a random component that

has mean over long period of time of zero but a variance of 2k T

B ^{γ . Thus, for an MD simu­}

lation that involves implicit solvation, LD can be included by a random sampled component

from a Gaussian distribution of mean zero and variance 2k T

B ^γ.

FIGURE 8.3  Multilayered solvent model. Schematic example of a hybrid molecular simula­

tion approach, here shown with a canonical segment of a Z-​DNA (see Chapter 2), which might

be simulated using either a QM ab initio method or classical (molecular mechanics) dynamics

simulations (MD) approach, or a hybrid of the two, while around it is a solvent shell in which

the trajectories of water molecules are simulated explicitly using classical MD, and around this

shell is an outer zone of implicit solvent in which continuum model is used to account for the

effects of water.