Chapter 8 – Theoretical Biophysics  333

The value of γ/​m =​ 1/​τ is a measure of the collision frequency between the biomolecule

atoms and water molecules, where τ is the mean time between collisions. For example, for

individual atoms in a protein, τ is in the range 25–​50 ps, whereas for water molecules, τ is

~80 ps. The limit of high γ values is an overdamped regime in which viscous forces dominate

over inertial forces. Here τ can for some biomolecule systems in water be as low as ~1 ps,

which is a diffusion (as opposed to stochastic) dominated limit called Brownian dynamics.

The equation of motion in this regime then reduces to the Smoluchowski diffusion equation,

which we will discuss later in this chapter in the section on reaction–​diffusion analysis.

One effect of applying a stochastic frictional drag term in LD is that this can be used to

slow down the motions of fast-​moving particles in the simulation and thus act as feedback

mechanism to clamp the particle speed range within certain limits. Since the system tem­

perature depends on particle speeds, this method thus equates to a Langevin thermostat (or

equivalently a barostat to maintain the pressure of the system). Similarly, other nonstochastic

thermostat algorithms can also be used, which all in effect include additional weak frictional

coupling constants in the equation of motion, including the Anderson, isokinetic/​Gaussian,

Nosé–​Hoover, and Berendsen thermostats.

8.2.8  COARSE-​GRAINED SIMULATION TOOLS

There are a range of coarse-​grained (CG) simulation approaches that, instead of probing the

exact coordinates of every single atom in the system independently, will pool together groups

of atoms as a rigid, or semirigid, structure, for example, as connected atoms of a single amino

acid residue in a protein, or coarser still of groups of residues in a single structural motif

in a protein. The forces experienced by the components of biological matter in these CG

simulations can also be significantly simplified versions that only approximate the underlying

QM potential energy. These reductions in model complexity are a compromise to achieving

computational tractability in the simulations and ultimately enable larger length and time

scales to be simulated at the expense of loss of fine detail in the structural makeup of the

simulated biological matter.

This coarse-​graining costs less computational time and enables longer simulation times to

be achieved, for example, time scales up to ~10⁻⁵ s can be simulated. However, again there is a

time scale gap since large molecular conformational changes under experimental conditions

can be much slower than this, perhaps lasting hundreds to thousands of microseconds.

Further coarse-​graining can allow access into these longer time scales, for example, by

pooling together atoms into functional structural motifs and modeling the connection

between the motifs with, in effect, simple springs, resulting in a simple harmonic potential

energy function.

Mesoscale models are at a higher length scale of coarse-​graining, which can be applied

to larger macromolecular structures. These, in effect, pool several atoms together to create

relatively large soft-​matter units characterized by relatively few material parameters, such

as mechanical stiffness, Young’s modulus, and the Poisson ratio. Mesoscale simulations can

model the behavior of macromolecular systems potentially over a time scale of seconds, but

clearly what they lack is the fine detail of information as to what happens at the level of specific

single molecules or atoms. However, in the same vein of hybrid QM/​MM simulations, hybrid

mesoscale/​CG approaches can combine elements of mesoscale simulations with smaller

length scale CG simulations, and similarly hybrid CG/​MM simulations can be performed,

with the strategy for all these hybrid methods that the finer length scale simulation tools

focus on just highly localized regions of a biomolecule, while the longer length scale simula­

tion tool generates peripheral information about the surrounding structures.

Hybrid QM/​MM approaches are particularly popular for investigating molecular docking

processes. that is, how well or not a small ligand molecule binds to a region of another larger

molecule. This process often utilizes relative simple scoring functions to generate rapid

estimates for the goodness of fit for a docked molecule to a putative binding site to facilitate

fast computational screening and is of specific interest in in silico drug design, that is, using

computational molecular simulations to develop new pharmaceutical chemicals.