Introduction to Molecular Dynamics: Theory and Applications in Biomolecular Modeling - Computational Modeling of Biological Systems - page 5

often of less interest or practical value to us than the macroscopic properties of
a system, which are the only properties that most experiments can measure. For
instance, knowing the exact locations of individual water molecules in a simulation
box may be less interesting or important than knowing the average rate at which
they are conducted by a channel protein across a lipid membrane [ 55 , 120 ].
The relationship between microscopic and macroscopic properties of a system is
the subject of statistical mechanics. On this topic, many excellent reference topics
are available [ 22 , 68 , 89 ], including some that offered discussions in the specific
context of computer simulations and biomolecular systems [ 5 , 34 , 38 ]. Below, we
will give a brief overview of some of the key concepts, and we encourage the readers
to the aforementioned topics for more information.

2.1 Microstates and the Ensemble Theory

A microscopic state of a system is specified by the positions and momenta of all
particles in the system. For a system with N particles, we may write its Hamiltonian
H as a sum of the kinetic energy K and the potential energy V , which are functions
of the Cartesian momentum p i and coordinate r i of each particle i, respectively:

r D . r 1 ; r 2 ;:::; r N /; (1)

p D . p 1 ; p 2 ;:::; p N /; (2)

H. r ; p / D K. p / C V. r /: (3)

Usually, the kinetic energy takes the familiar quadratic form:

N

X

1
2m i .p ix C p iy C p i z /;

K D (4)

i D 1

where m i is the mass of particle i,andp ix , p iy , p i z are the x, y,and z components
of its momentum p i . The potential energy V has a much more complicated form and
will be discussed in more detail later. For now, it suffices to say that once the form
of V is determined, the time evolution of the system, governed by the Hamiltonian
H, can be determined by solving the equations of motion in a MD simulation. If
we think of the positions and momenta of all particles in the system as coordinates
in a 6N-dimension space, which we refer to as the phase space, then at any given
time, the system corresponds to a point in this multidimensional space. The time
evolution of the system, thereby, corresponds to a trajectory in the phase space.
As mentioned earlier, we are interested in calculating certain macroscopic
properties of the system. Instead of following the trajectory of a single system in the
phase space, the conventional approach used in statistical mechanics is to consider,
at any given time, a collection of systems with the same macroscopic properties,