Introduction to Molecular Dynamics: Theory and Applications in Biomolecular Modeling - Computational Modeling of Biological Systems - page 8

3 Molecular Dynamics: The Theory

Let's go back to the Hamiltonian H defined in ( 3 ), which governs the time evolution
of a system. We have shown that the kinetic part of H usually has a simple
quadratic form ( 4 ). Now it's time to look at the more complicated potential part
of H. Unfortunately, the actual form of the potential energy V is so complicated
that even with the power of modern computers, some approximations are needed
before we can calculate it efficiently. In this section, we will first go through these
approximations, and then discuss how to use MD simulations to generate the desired
phase space distribution of a statistical ensemble.

3.1 The Force Field

In MD, the specific form of the potential energy function V is given as a force field,
where V is broken down into terms characterizing different types of interactions in
a bimolecular system. Examples of some of the most commonly used force fields
are AMBER [ 54 , 117 ], CHARMM [ 69 , 79 - 81 ], GROMOS [ 92 , 102 ], and OPLS [ 29 ,
63 , 96 ]. Although these force fields were initially developed by different groups,
most of them have similar functional forms, i.e., the potential energy V is divided
into bonded and nonbonded terms, the former of which includes the bond, angle,
dihedral, and improper interaction terms, while the latter includes the Van der Waals
(vdW) and electrostatic interaction terms. As an example, energy functions from the
CHARMM force field are shown below:

V D V bond C V angle C V dihedral C V improper C V vdw C V elec ; (12)

V bond D K b .b b 0 / 2 ;

(13)

V angle D K . 0 / 2 ; (14)

V dihedral D K 1 C cos.n ı/ ;

(15)

V improper D K . 0 / 2 ;

(16)

V vdw D " h R ij
r ij

12

2 R ij
r ij

6

i ;

(17)

q i q j
4" 0 r ij :

V elec D (18)

In Fig. 2 , we have given the schematic representation of each energy term listed
above, along with a plot showing the corresponding energy value for certain selected
atom types. While the bond and angle terms are rather straightforward to understand,
the rest of the terms may require some explanation. Mathematically, a dihedral and
an improper both involve four atoms, and are defined as the angle between the