320 8.2  Molecular Simulation Methods

offer a good compromise enabling biomolecule systems containing typically 10⁴–​10⁵ atoms to

be simulated for ~10–​100 ns. The reduction from a quantum to a classical description entails

two key assumptions. First, electron movement is significantly faster than that of atomic

nuclei such that we assume they can change their relative position instantaneously. This is

called the “Born–​Oppenheimer approximation,” which can be summarized as the total wave

function being the product of the orthogonal wave functions due to nuclei and electrons

separately:

(8.7)

ψ

ψ

ψ

total ^{nuclei electrons}

(nuclei

(electrons

,

)

)

(

) =

The second assumption is that atomic nuclei are treated as point particles of much greater

mass than the electrons that obey classical Newtonian dynamics. These approximations lead

to a unique potential energy function Utotal due to the relative positions of electrons to nuclei.

Here, the force F on a molecule is found from

(8.8)

F

U

r

= −∇

( )

total



Utotal is the total potential energy function in the vicinity of each molecule summed from

all relevant repulsive and attractive force sources experienced by each, whose position is

denoted by the vector r. A parameter sometimes used to determine thermodynamic prop­

erties such as free energy is the potential of mean force (PMF) (not to be confused with the

proton motive force; see Chapter 2), which is the potential energy that results in the average

force calculated over all possible interactions between atoms in the system.

In practice, most classical MD simulations use relatively simple predefined potentials.

To model the effects of chemical bonding between atoms, empirical potentials are used.

These consist of the summation of independent potential energy functions associated with

bonding forces between atoms, which include the covalent bond strength, bond angles, and

bond dihedral potentials (a dihedral, or torsion angle, is the angle between two intersecting

planes generated from the relative atomic position vectors). Nonbonding potential energy

contributions come typically from van der Waals (vdW) and electrostatic forces. Empirical

potentials are limited approximations to QM effects. They contain several free parameters

(including equilibrium bond lengths, angles and dihedrals, vdW potential parameters, and

atomic charge) that can be optimized either by fitting to QM simulations or from separate

experimental biophysical measurements.

The simplest nonbonding empirical potentials consider just pairwise interactions between

nearest-​neighbor atoms in a biological system. The most commonly applied nonbonding

empirical potential in MD simulations is the Lennard–​Jones potential ULJ (see Chapter 2), a

version of which is given in Equation 2.10. It is often also written in the form

(8.9)

U

r

r

r

r

r

r

LJ

m

m

=



^



^



^



^



^



^=



^



^



^



^









4

12

6

12

6

ε

σ

σ

ε





where

r is the distance between the two interacting atoms

ε is the depth of the potential well

σ is the interatomic distance that results in ULJ =​ 0

rm is the interatomic distance at which the potential energy is a minimum (and thus the

force F =​ 0) given by

(8.10)

rm =

(

)

2

1 22

1 6

/

.

σ